Difference between revisions of "Find a Number Problems"

Latest revision as of 14:40, 12 July 2024

Find a Perfect Number

[Expand]perfect number

Find One Number

\scriptstyle37a=49

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\scriptstyle9a=1

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\scriptstyle13\sdot7\sdot3\sdot a=3

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\scriptstyle\frac{7a}{9}=49

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\scriptstyle\frac{a}{5}\sdot8=23

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\scriptstyle\frac{\frac{1}{5}a}{\sqrt{a}}=7

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\scriptstyle\frac{3}{a}=5+\frac{7}{b}

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\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=10

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\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=10+\frac{1}{2}

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\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=20

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\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=6

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\scriptstyle\left(\frac{1}{4}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=5

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\scriptstyle a+\left(\frac{1}{3}\sdot a\right)+\left(\frac{2}{5}\sdot a\right)=81

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\scriptstyle a+\left[\left(2+\frac{2}{3}\right)\sdot a\right]=11

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\scriptstyle\left(\frac{1}{2}\sdot a\right)+\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)-5=24

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\scriptstyle a+\frac{1}{2}\sdot\left(\frac{1}{4}\sdot a\right)+\left(\frac{3}{4}\sdot a\right)+\frac{1}{4}\sdot\left(\frac{1}{4}\sdot a\right)=2

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\scriptstyle a+\frac{1}{2}a+\frac{1}{5}a+\frac{1}{7}a=2

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\scriptstyle a+\frac{1}{2}\sdot\frac{1}{2}a+\frac{1}{3}\sdot\frac{1}{3}a+\frac{1}{5}\sdot\frac{1}{5}a=7

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\scriptstyle\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]+\left[\frac{3}{4}\sdot\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]\right]=13

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\scriptstyle a-\left(\frac{1}{3}\sdot a\right)-\left(\frac{1}{4}\sdot a\right)=29

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\scriptstyle\left(\frac{1}{2}a+\frac{1}{3}a\right)-\left(\frac{1}{4}a+\frac{1}{5}a\right)=4

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\scriptstyle a-\left(\frac{1}{3}\sdot a\right)=\sqrt{a}

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\scriptstyle\left[a-\left(\frac{2}{3}\sdot a\right)\right]^2=20

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\scriptstyle\left[a\sdot\left(\frac{2}{3}\sdot a\right)\right]^2=36

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\scriptstyle a^2\sdot\left(\frac{3}{4}\sdot a^2\right)=6a

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\scriptstyle\frac{2}{5}a^2=a

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\scriptstyle a+28=2a^2

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\scriptstyle a^2=4\sdot\left(a+8\right)

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\scriptstyle a^2=\left(3+\frac{10}{13}\right)\sdot\left(a+6\right)

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\scriptstyle a^2\sdot a=12a+10a^2

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\scriptstyle\sqrt{9a}=12

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\scriptstyle\sqrt{a}+\sqrt{a-\sqrt{a}}=3

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\scriptstyle\left(a\sdot\frac{2}{3}a\right)^2=\sqrt{50}

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\scriptstyle\left(a\sdot\frac{2}{3}a\right)\sdot a=a\sdot\sqrt{8}

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\scriptstyle\frac{3}{7}a+\frac{4}{5}a=\frac{2}{3}a+\frac{1}{4}a+20

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\scriptstyle\frac{1}{4}\sdot a=\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]=\frac{1}{2}\sdot\left[\left[a-\left(\frac{1}{4}\sdot a\right)\right]-\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]\right]

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\scriptstyle\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]-\left[\frac{1}{2}\sdot\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]+\frac{1}{2}\right]=1

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\scriptstyle\frac{3a}{4}:4=4:6

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\scriptstyle2:3=4:a

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\scriptstyle4=\frac{1}{3}\sdot15\longrightarrow a=\frac{1}{5}\sdot25

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\scriptstyle a^2=40+\frac{41}{b}

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\scriptstyle\frac{4a}{5}=6+\frac{3}{5}

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\scriptstyle4a\bmod5=1

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\scriptstyle4a\bmod6=1

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\scriptstyle a\bmod3=a\bmod5=a\bmod7=1

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\scriptstyle a\bmod3=2;a\bmod5=4;a\bmod7=6

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\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=0

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\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=0

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\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=0

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\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=1;a\bmod6=1;a\bmod7=0

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\scriptstyle a\bmod1=0;a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=1

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\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=1;a\bmod6=5;a\bmod7=1;a\bmod8=7;a\bmod9=8;a\bmod10=1

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\scriptstyle a\bmod2=0;a\bmod3=2;a\bmod4=0;a\bmod5=0;a\bmod6=2;a\bmod3=1;a\bmod8=4;a\bmod9=5;a\bmod10=0

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\scriptstyle a^2+19=b^2

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\scriptstyle a^2-15=b^2

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\scriptstyle\begin{cases}\scriptstyle a+b=n^2\\\scriptstyle a-b=m^2\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a+4=n^2\\\scriptstyle a-4=m^2\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a^2-5a=n^2\\\scriptstyle a^2+7a=m^2\end{cases}

[Expand]repeated subtraction

Find Two Numbers

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\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle2\sdot\left(a-1\right)=b+1\end{cases}

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\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle20\sdot\left(a-1\right)=b+1\end{cases}

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\scriptstyle\begin{cases}\scriptstyle2\sdot\left(b-1\right)=a+1\\\scriptstyle3\sdot\left(a-1\right)=b+1\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a^2=b-5\\\scriptstyle a^2+b^2=100\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a^2=2b+8\\\scriptstyle a^2+b^2=80\end{cases}

b=2a

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\scriptstyle\begin{cases}\scriptstyle b=2a\\\scriptstyle a^2\sdot b=9b\end{cases}

a:b=2:3

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle4a+4b=\sqrt{100}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=\sqrt{12}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(3a+4b\right)\sdot\sqrt{5}=40\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=100\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=2b\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=b^2\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=b\sdot\sqrt{8}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a^2\sdot b=b\sdot\sqrt{b}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle5a+7b=16+\sqrt{8}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4\sqrt{b}=30\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=20+\sqrt{8}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot b=100+\sqrt{8}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a^2\sdot b\right)+\sqrt{a^2\sdot b}=342\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=20+\sqrt{30}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2+\left(a^2\sdot\sqrt{4}\right)=4+\frac{1}{4}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4b=\sqrt[3]{216}\end{cases}

a:b=3:4

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=288\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)\sdot\left(a+b\right)=a^2\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=\sqrt{20+\frac{1}{4}}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle8a=6\sqrt{b}\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot\sqrt{8}\right)+b^2=48\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)^2+27=9b^2\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\sqrt[3]{2a+3b}=8\end{cases}

a:b=4:5

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\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot b\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle a\sdot b=b\sdot\sqrt{8}\end{cases}

a:b=3:5

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\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left[\left(a\sdot\sqrt{a}\right)+\left(b\sdot\sqrt{b}\right)\right]\sdot\sqrt{8}=100\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=b\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a^2\sdot b=b^2\sdot\sqrt{8}\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a\sdot b=\sqrt[3]{729}\end{cases}

a:b=2:5

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\scriptstyle\begin{cases}\scriptstyle a:b=2:5\\\scriptstyle a\sdot b=8\sqrt{b}\end{cases}

a:b=1:4

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\scriptstyle\begin{cases}\scriptstyle a:b=1:4\\\scriptstyle\left(a\sdot b\right)\sdot a=\sqrt[3]{216}\end{cases}

a:b=1:3

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\scriptstyle\begin{cases}\scriptstyle a:b=1:3\\\scriptstyle\sqrt[3]{a}\sdot\sqrt[3]{b}=100\end{cases}

a:b=5:7

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\scriptstyle\begin{cases}\scriptstyle a:b=5:7\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot\sqrt{8}\end{cases}

8:a:b:27

[Expand]

\scriptstyle8:a:b:27

Find Three Numbers

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\scriptstyle\begin{cases}\scriptstyle a+b=c\\\scriptstyle a+c=100b\end{cases}

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\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle b:c=3:4\\\scriptstyle\left(a^2\sdot a\right)+b^2=\left(12+\frac{1}{2}\right)\sdot c\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=3+\frac{2}{5}+\frac{1}{7}\\\scriptstyle\left(a+c\right):b=7+\frac{2}{3}+\frac{1}{4}\\\scriptstyle b=30\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=\frac{3}{5}+\frac{1}{6}\\\scriptstyle\left(a+c\right):b=2+\frac{1}{3}\\\scriptstyle a=20\end{cases}

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\scriptstyle\begin{cases}\scriptstyle b:\left(c-a\right)=3+\frac{1}{3}\\\scriptstyle c:\left(b-a\right)=6\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):\left(c-a\right)=4+\frac{1}{2}\\\scriptstyle\left(a+c\right):\left(b-a\right)=5\end{cases}

[Expand]

\scriptstyle\begin{cases}\scriptstyle a+\frac{1}{4}\sdot\left(b+c\right)=b+\frac{1}{6}\sdot\left(a+c\right)=c+\frac{1}{9}\sdot\left(a+b\right)\\\scriptstyle b=20\end{cases}

Category	Comment	Link	Annotated text
word problem/find a number	a+¼(b+c)=b+⅙(a+c)=c+⅑(a+b);b=20	ספר_מעשה_חושב#sryH	Example: the first with a quarter of the rest is equal to the second with a sixth of the rest and is also equal to the third with a ninth of the rest. ::The first is added to the greatest part of the rest and the third is added to the smallest part of the rest. ::The second number is 20. ::We wish to know: how much is each of the rest? :: $\scriptstyle\begin{cases}\scriptstyle a+\frac{1}{4}\sdot\left(b+c\right)=b+\frac{1}{6}\sdot\left(a+c\right)=c+\frac{1}{9}\sdot\left(a+b\right)\\\scriptstyle b=20\end{cases}$ והמשל שיהיה הראשון עם רביעית הנשארים שוה אל השני עם ששית הנשארים והוא גם כן שוה אל השלישי עם תשיעית הנשארים וכבר התבאר לך במאמר הראשון מזה הספר שסדר אלו המספרים השלשה הוא על זאת ההדרגה אשר סדרנו רצוני שהראשון הוא אשר יתחבר עמו חלק יותר גדול מהנשארים והשלישי הוא אשר יתחבר עמו חלק יותר קטן מהנשארים והזכרתי לך זה למען לא תתבלבל בסדורם ויהיה המספר השני מהם כ‫' ורצינו לדעת כמה כל אחד מהנשארים
word problem/find a number	a+¼(b+c)=b+⅙(a+c)=c+⅑(a+b);b=20	ספר_דיני_ממונות#HxDL	62) Question: the first with a quarter of the rest is equal to the second with a sixth of the rest and is also equal to the third with a ninth of the rest. :It has already been explained to you in the first section of this book that the order of these three numbers is according to this arrangement, meaning that the first is added to the greatest part of the rest and the [third] is added to the smallest part of the rest. I remind you this so that you will not be confused by the explanations. :The second number is 20. :We wish to know: how much is each of the rest? סב) שאלה שיהיה הראשון עם רביעית הנשארים שוה אל השני עם שישית הנשארי' והוא ג"כ שוה אל השלישי עם תשיעית הנשארים כבר התבאר לך במאמ' הראשון מזה הספר שסדור אלו המספרים הג' הוא על זה הסדר אשר סדרוה רצוני שהראשון הוא אשר יתחבר המתחלק יותר גדול מן הנשארים והל' הוא אשר יתחבר עמו יותר קטן והנשארים והזכרתי זה לך למען לא תבלבל בביאורים והנה יהיה המספר השני מהם כ‫' ורצינו לדעת כמה כל א' מהנשארים

Three Proportional Numbers

[Expand]three proportional numbers

[Expand]

\scriptstyle\begin{cases}\scriptstyle a:b=b:c\\\scriptstyle a\sdot b\sdot c=1000\end{cases}

Four Proportional Numbers

[Expand]four proportional numbers

Five Proportional Numbers

[Expand]five proportional numbers

Six Proportional Numbers

[Expand]six proportional numbers

Category	Comment	Link	Annotated text
word problem/find a number	six proportional numbers	ספר_דיני_ממונות#Wrir	For example: we have six proportional numbers, the first is 2 and the sixth is 64. ::We wish to know which is the second. דמיון זה יש לנו ו' {{#annot:term\|994,1277\|VUeX}}מספרים מתיחסים{{#annotend:VUeX}} הראשון ב' והו' ס"ד נרצה לדעת מהו השני
word problem/find a number	six proportional numbers	ספר_דיני_ממונות#FvyI	Example: the first is 2-[eighths] and the sixth is 256 integers. דמיון הראשון ב' שלישיות והו' רנ"ו שלמים
word problem/find a number	six proportional numbers	ספר_דיני_ממונות#fNcE	Example: we have six proportional numbers, the first of which is 2 and the sixth is 64. דמיון יש לנו ו' מספרים מתיחסים שהראשון ב' והו' ס"ד
word problem/find a number	six proportional numbers	ספר_דיני_ממונות#Yfpk	111) Question: if you want to find a proportional number: :For example: we have six proportional numbers, i.e. the ratio of the sixth to the fifth is the same as the ratio of the fifth to the fourth, and the same as the ratio of the fourth to the third, and the same as the ratio of the third to the second, and the same as the ratio of the second to the first. :The first number is 2. :The sixth is 2048. :We wish to know how much is the second? קיא)‫MS L: קט שאלה אם תרצה למצא {{#annot:term\|994,2263\|2xtB}}מספר יחסיי{{#annotend:2xtB}} הנקרא בלשונם {{#annot:term\|994\|DhLM}}נומירו פרופורציאונלי{{#annotend:DhLM}} דמיון זה יש לנו ו' {{#annot:term\|994,2263\|jjwV}}מספרים יחסיים{{#annotend:jjwV}} ר"ל שיחס הו' אל הה' כיחס הה' אל הד' וכיחס הד' אל הג' וכיחס הג' אל הב' וכיחס הב' אל הא‫' והמספר הא' ב‫' והו' ב' אלפים ומ"ח ונרצה לדעת מהו השני

Seven Proportional Numbers

[Expand]seven proportional numbers

Difference between revisions of "Find a Number Problems"

Latest revision as of 14:40, 12 July 2024

Contents

Find a Perfect Number

Find One Number

Find Two Numbers

b=2a

a:b=2:3

a:b=3:4

a:b=4:5

a:b=3:5

a:b=2:5

a:b=1:4

a:b=1:3

a:b=5:7

8:a:b:27

Find Three Numbers

Three Proportional Numbers

Four Proportional Numbers

Five Proportional Numbers

Six Proportional Numbers

Seven Proportional Numbers

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@@ Line 1: / Line 1: @@
+== Find a Perfect Number ==
+<div class="mw-collapsible mw-collapsed">perfect number<div class="mw-collapsible-content">
 {{#annotask:
 	[[category: #find a number]]
-	[[comment: repeated subtraction]]
+	[[comment: perfect number]]
-}}
+}}</div></div>
 == Find One Number ==
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle37a=49</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 37a=49]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle9a=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 9a=1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle13\sdot7\sdot3\sdot a=3</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 13·7·3a=3]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{7a}{9}=49</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 7a/9=45]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{a}{5}\sdot8=23</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (5/a)·8=23]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{\frac{1}{5}a}{\sqrt{a}}=7</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ⅕a/√a=7]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{a}=5+\frac{7}{b}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 3/a=5+7/b]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=10</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ⅓a+⅕a=10]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=10+\frac{1}{2}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ⅓a+¼a=10½]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=20</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ⅓a+¼a=20]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)=6</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ⅓a+¼a=6]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{4}\sdot a\right)+\left(\frac{1}{5}\sdot a\right)=5</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ¼a+⅕a=5]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\left(\frac{1}{3}\sdot a\right)+\left(\frac{2}{5}\sdot a\right)=81</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+⅓a+⅖a=81]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\left[\left(2+\frac{2}{3}\right)\sdot a\right]=11</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+2⅔a=11]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{2}\sdot a\right)+\left(\frac{1}{3}\sdot a\right)+\left(\frac{1}{4}\sdot a\right)-5=24</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ½a+⅓a+¼a-5=24]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\frac{1}{2}\sdot\left(\frac{1}{4}\sdot a\right)+\left(\frac{3}{4}\sdot a\right)+\frac{1}{4}\sdot\left(\frac{1}{4}\sdot a\right)=2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+½¼a+¾a+¼¼a=2]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\frac{1}{2}a+\frac{1}{5}a+\frac{1}{7}a=2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+½a+⅕a+⅐a=2]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+\frac{1}{2}\sdot\frac{1}{2}a+\frac{1}{3}\sdot\frac{1}{3}a+\frac{1}{5}\sdot\frac{1}{5}a=7</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+½½a+⅓⅓a+⅕⅕a=7]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]+\left[\frac{3}{4}\sdot\left(a+\frac{1}{3}a\right)+\left[\frac{2}{5}\sdot\left(a+\frac{1}{3}a\right)\right]\right]=13</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+⅓a+⅖(a+⅓a)+¾(a+⅓a+⅖(a+⅓a))=13]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a-\left(\frac{1}{3}\sdot a\right)-\left(\frac{1}{4}\sdot a\right)=29</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a-⅓a-¼a=29]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(\frac{1}{2}a+\frac{1}{3}a\right)-\left(\frac{1}{4}a+\frac{1}{5}a\right)=4</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (½a+⅓a)-(¼a+⅕a)=4]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a-\left(\frac{1}{3}\sdot a\right)=\sqrt{a}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a-⅓a=√a]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[a-\left(\frac{2}{3}\sdot a\right)\right]^2=20</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (a-⅓a)²=20]]
+}}</div></div>
+<br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[a\sdot\left(\frac{2}{3}\sdot a\right)\right]^2=36</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find one number]]
+	[[category: #find a number]]
 	[[comment: (a·⅔a)²=36]]
 }}</div></div>
@@ Line 14: / Line 156: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2\sdot\left(\frac{3}{4}\sdot a^2\right)=6a</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find one number]]
+	[[category: #find a number]]
 	[[comment: a²·¾a²=6a]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{2}{5}a^2=a</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ⅖a²=a]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a+28=2a^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+28=2a²]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2=4\sdot\left(a+8\right)</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²=4(a+8)]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2=\left(3+\frac{10}{13}\right)\sdot\left(a+6\right)</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²=(3+¹⁰/₁₃)(a+6)]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2\sdot a=12a+10a^2</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find one number]]
+	[[category: #find a number]]
 	[[comment: a²·a=12a+10a²]]
 }}</div></div>
@@ Line 26: / Line 192: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{9a}=12</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find one number]]
+	[[category: #find a number]]
 	[[comment: √(9a)=12]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\sqrt{a}+\sqrt{a-\sqrt{a}}=3</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: √a+√(a-√a)=3]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a\sdot\frac{2}{3}a\right)^2=\sqrt{50}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find one number]]
+	[[category: #find a number]]
 	[[comment: (a·⅔a)²=√50]]
 }}</div></div>
@@ Line 38: / Line 210: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left(a\sdot\frac{2}{3}a\right)\sdot a=a\sdot\sqrt{8}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find one number]]
+	[[category: #find a number]]
 	[[comment: (a·⅔a)·a=a√8]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3}{7}a+\frac{4}{5}a=\frac{2}{3}a+\frac{1}{4}a+20</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ³/₇a+⁴/₅a=⅔a+¼a+20]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{1}{4}\sdot a=\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]=\frac{1}{2}\sdot\left[\left[a-\left(\frac{1}{4}\sdot a\right)\right]-\frac{1}{3}\left[a-\left(\frac{1}{4}\sdot a\right)\right]\right]</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: ¼a=⅓(a-¼a)=½((a-¼a)-⅓(a-¼a))]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]-\left[\frac{1}{2}\sdot\left[\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+\frac{1}{2}\right)\right]\right]+\frac{1}{2}\right]\right]+\frac{1}{2}\right]=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a-(½a+½)-(½(a-(½a+½))+½)-(½(a-(½a+½)-(½(a-(½a+½))+½))+½)]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{3a}{4}:4=4:6</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (3a/4)/4=4/6]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle2:3=4:a</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 2/3=4/a]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4=\frac{1}{3}\sdot15\longrightarrow a=\frac{1}{5}\sdot25</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 4=⅓·15→a=⅕·25]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2=40+\frac{41}{b}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²=40+41\b]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\frac{4a}{5}=6+\frac{3}{5}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 4a/5=6⅗]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4a\bmod5=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 4a(mod)5=1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle4a\bmod6=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 4a(mod)6=1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod3=a\bmod5=a\bmod7=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod)3=a(mod5)=a(mod7)=1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod3=2;a\bmod5=4;a\bmod7=6</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod)3=2,a(mod5)=4,a(mod7)=6]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=0</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod2)=1,a(mod3)=1,a(mod)4=1,a(mod5)=0]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=0</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=0]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=0</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=4,a(mod6)=5,a(mod7)=0]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=1;a\bmod4=1;a\bmod5=1;a\bmod6=1;a\bmod7=0</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod2)=1,a(mod3)=1,a(mod)4=1,a(mod5)=1,a(mod6)=1,a(mod7)=0]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod1=0;a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=4;a\bmod6=5;a\bmod7=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod)1=0,a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=4,a(mod6)=5,a(mod7)=1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=1;a\bmod3=2;a\bmod4=3;a\bmod5=1;a\bmod6=5;a\bmod7=1;a\bmod8=7;a\bmod9=8;a\bmod10=1</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod2)=1,a(mod3)=2,a(mod)4=3,a(mod5)=1,a(mod6)=5,a(mod7)=1,a(mod8)=7,a(mod9)=8,a(mod10)=1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a\bmod2=0;a\bmod3=2;a\bmod4=0;a\bmod5=0;a\bmod6=2;a\bmod3=1;a\bmod8=4;a\bmod9=5;a\bmod10=0</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a(mod2)=0,a(mod3)=2,a(mod)4=0,a(mod5)=0,a(mod6)=2,a(mod7)=3,a(mod8)=4,a(mod9)=5,a(mod10)=0]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2+19=b^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²+19=b²]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle a^2-15=b^2</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²-15=b²]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+b=n^2\\\scriptstyle a-b=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+b=n², a-b=m²]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+4=n^2\\\scriptstyle a-4=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+4=n², a-4=m²]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2-5a=n^2\\\scriptstyle a^2+7a=m^2\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²-5a=n², a²+7a=m²]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed">repeated subtraction<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: repeated subtraction]]
 }}</div></div>
 == Find Two Numbers ==
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle2\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: b-1=a+1;2(a-1)=b+1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b-1=a+1\\\scriptstyle20\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: b-1=a+1;20(a-1)=b+1]]
+}}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle2\sdot\left(b-1\right)=a+1\\\scriptstyle3\sdot\left(a-1\right)=b+1\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 2(b-1)=a+1;3(a-1)=b+1]]
+}}</div></div>
+<br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2=b-5\\\scriptstyle a^2+b^2=100\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #find two numbers]]
+	[[category: #find a number]]
 	[[comment: a²=b-5;a²+b²=100]]
 }}</div></div>
+<br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a^2=2b+8\\\scriptstyle a^2+b^2=80\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a²=2b+8;a²+b²=80]]
+}}</div></div>
 === b=2a ===
@@ Line 54: / Line 405: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b=2a\\\scriptstyle a^2\sdot b=9b\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #one double the other]]
+	[[category: #find a number]]
-	[[comment: a²b=9b]]
+	[[comment: b=2a;a²b=9b]]
 }}</div></div>
@@ Line 62: / Line 413: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle4a+4b=\sqrt{100}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: 4a+4b=√100]]
+	[[comment: a/b=⅔;4a+4b=√100]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=\sqrt{12}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: ab=√12]]
+	[[comment: a/b=⅔;ab=√12]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(3a+4b\right)\sdot\sqrt{5}=40\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: (3a+4b)√5=40]]
+	[[comment: a/b=⅔;(3a+4b)√5=40]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=100\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: ab√8=100]]
+	[[comment: a/b=⅔;ab√8=100]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=2b\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: a√a=2b]]
+	[[comment: a/b=⅔;a√a=2b]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot\sqrt{a}=b^2\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: a√a=b²]]
+	[[comment: a/b=⅔;a√a=b²]]
-}}}}</div></div>
+}}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=b\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: (ab)²=b√8]]
+	[[comment: a/b=⅔;(ab)²=b√8]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a^2\sdot b=b\sdot\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: a²b=b√b]]
+	[[comment: a/b=⅔;a²b=b√b]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle5a+7b=16+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: 5a+7b=16+√8]]
+	[[comment: a/b=⅔;5a+7b=16+√8]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4\sqrt{b}=30\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: 3a+4√b=30]]
+	[[comment: a/b=⅔;3a+4√b=30]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle a\sdot b=20+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: ab=20+√8]]
+	[[comment: a/b=⅔;ab=20+√8]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)\sdot b=100+\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: (ab)b=100+\sqrt{8}]]
+	[[comment: a/b=⅔;(ab)b=100+\sqrt{8}]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a^2\sdot b\right)+\sqrt{a^2\sdot b}=342\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: a²b+\sqrt{a²b}=342]]
+	[[comment: a/b=⅔;a²b+\sqrt{a²b}=342]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2=20+\sqrt{30}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: (ab)²=20+√30]]
+	[[comment: a/b=⅔;(ab)²=20+√30]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle\left(a\sdot b\right)^2+\left(a^2\sdot\sqrt{4}\right)=4+\frac{1}{4}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: (ab)²+a²√4=4¼]]
+	[[comment: a/b=⅔;(ab)²+a²√4=4¼]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle3a+4b=\sqrt[3]{216}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅔]]
+	[[category: #find a number]]
-	[[comment: 3a+4b=³√216]]
+	[[comment: a/b=⅔;3a+4b=³√216]]
 }}</div></div>
 <br>
@@ Line 161: / Line 512: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=288\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: a²b=288]]
+	[[comment: a/b=¾;a²b=288]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)\sdot\left(a+b\right)=a^2\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: (ab)·(a+b)=a²]]
+	[[comment: a/b=¾;(ab)·(a+b)=a²]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle a^2\sdot b=\sqrt{20+\frac{1}{4}}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: a²b=√(20¼)]]
+	[[comment: a/b=¾;a²b=√(20¼)]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle8a=6\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: 8a=6√b]]
+	[[comment: a/b=¾;8a=6√b]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot\sqrt{8}\right)+b^2=48\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: a√8+b²=48]]
+	[[comment: a/b=¾;a√8+b²=48]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\left(a\sdot b\right)^2+27=9b^2\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: (ab)²+27=9b²]]
+	[[comment: a/b=¾;(ab)²+27=9b²]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:4\\\scriptstyle\sqrt[3]{2a+3b}=8\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¾]]
+	[[category: #find a number]]
-	[[comment: ³√(2a+3b)=8]]
+	[[comment: a/b=¾;³√(2a+3b)=8]]
 }}</div></div>
 <br>
@@ Line 206: / Line 557: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot b\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅘]]
+	[[category: #find a number]]
-	[[comment: (ab)²=a²b]]
+	[[comment: a/b=⅘;(ab)²=a²b]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=4:5\\\scriptstyle a\sdot b=b\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅘]]
+	[[category: #find a number]]
-	[[comment: ab=b√8]]
+	[[comment: a/b=⅘;ab=b√8]]
 }}</div></div>
@@ Line 220: / Line 571: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)^2=a^2\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅗]]
+	[[category: #find a number]]
-	[[comment: (ab)²=a²]]
+	[[comment: a/b=⅗;(ab)²=a²]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left[\left(a\sdot\sqrt{a}\right)+\left(b\sdot\sqrt{b}\right)\right]\sdot\sqrt{8}=100\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅗]]
+	[[category: #find a number]]
-	[[comment: (a√a+b√b)√8=100]]
+	[[comment: a/b=⅗;(a√a+b√b)√8=100]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle\left(a\sdot b\right)\sdot\sqrt{8}=b\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅗]]
+	[[category: #find a number]]
-	[[comment: ab√8=b]]
+	[[comment: a/b=⅗;ab√8=b]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a^2\sdot b=b^2\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅗]]
+	[[category: #find a number]]
-	[[comment: a²b=b²√8]]
+	[[comment: a/b=⅗;a²b=b²√8]]
 }}</div></div>
 <br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=3:5\\\scriptstyle a\sdot b=\sqrt[3]{729}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅗]]
+	[[category: #find a number]]
-	[[comment: ab=³√729]]
+	[[comment: a/b=⅗;ab=³√729]]
 }}</div></div>
 <br>
 === a:b=2:5 ===
@@ Line 254: / Line 604: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:5\\\scriptstyle a\sdot b=8\sqrt{b}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅖]]
+	[[category: #find a number]]
-	[[comment: ab=8√b]]
+	[[comment: a/b=⅖;ab=8√b]]
 }}</div></div>
 <br>
 === a:b=1:4 ===
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=1:4\\\scriptstyle\left(a\sdot b\right)\sdot a=\sqrt[3]{216}\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;¼]]
+	[[category: #find a number]]
-	[[comment: aba=³√216]]
+	[[comment: a/b=¼;aba=³√216]]
 }}</div></div>
 <br>
@@ Line 271: / Line 622: @@
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=1:3\\\scriptstyle\sqrt[3]{a}\sdot\sqrt[3]{b}=100\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b;⅓]]
+	[[category: #find a number]]
-	[[comment: ³√a³√b=100]]
+	[[comment: a/b=⅓;³√a³√b=100]]
+}}</div></div>
+<br>
+=== a:b=5:7 ===
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=5:7\\\scriptstyle\left(a\sdot b\right)^2=a^2\sdot\sqrt{8}\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a/b=⁵/₇;(ab)²=a²√8]]
+}}</div></div>
+<br>
+=== 8:a:b:27 ===
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle8:a:b:27</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: 8/a/b/27]]
 }}</div></div>
 <br>
 == Find Three Numbers ==
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+b=c\\\scriptstyle a+c=100b\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+b=c;a+c=100b]]
+}}</div></div><br>
 <div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=2:3\\\scriptstyle b:c=3:4\\\scriptstyle\left(a^2\sdot a\right)+b^2=\left(12+\frac{1}{2}\right)\sdot c\end{cases}</math><div class="mw-collapsible-content">
 {{#annotask:
-	[[category: #a/b/c;2/3/4]]
+	[[category: #find a number]]
 	[[comment: (a²·a)+b²=12½c]]
-}}</div></div>
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=3+\frac{2}{5}+\frac{1}{7}\\\scriptstyle\left(a+c\right):b=7+\frac{2}{3}+\frac{1}{4}\\\scriptstyle b=30\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (a+b)\c=3⅖⅐;(a+c)\b=7⅔¼;b=30]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=\frac{3}{5}+\frac{1}{6}\\\scriptstyle\left(a+c\right):b=2+\frac{1}{3}\\\scriptstyle a=20\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (a+b)/c=⅗+⅙;(a+c)/b=2⅓;a=20]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle b:\left(c-a\right)=3+\frac{1}{3}\\\scriptstyle c:\left(b-a\right)=6\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: b/(c-a)=3⅓;c/(b-a)=6]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):\left(c-a\right)=4+\frac{1}{2}\\\scriptstyle\left(a+c\right):\left(b-a\right)=5\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: (a+b)/(c-a)=4½;(a+c)/(b-a)=5]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a+\frac{1}{4}\sdot\left(b+c\right)=b+\frac{1}{6}\sdot\left(a+c\right)=c+\frac{1}{9}\sdot\left(a+b\right)\\\scriptstyle b=20\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a+¼(b+c)=b+⅙(a+c)=c+⅑(a+b);b=20]]
+}}</div></div><br>
+== Three Proportional Numbers==
+<div class="mw-collapsible mw-collapsed">three proportional numbers<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: three proportional numbers]]
+}}</div></div><br>
+<div class="mw-collapsible mw-collapsed"><math>\scriptstyle\begin{cases}\scriptstyle a:b=b:c\\\scriptstyle a\sdot b\sdot c=1000\end{cases}</math><div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: a·b·c=1000;a\b\c]]
+}}</div></div><br>
+== Four Proportional Numbers==
+<div class="mw-collapsible mw-collapsed">four proportional numbers<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: four proportional numbers]]
+}}</div></div><br>
+== Five Proportional Numbers ==
+<div class="mw-collapsible mw-collapsed">five proportional numbers<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: five proportional numbers]]
+}}</div></div><br>
+== Six Proportional Numbers ==
+<div class="mw-collapsible mw-collapsed">six proportional numbers<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: six proportional numbers]]
+}}</div></div><br>
+== Seven Proportional Numbers ==
+<div class="mw-collapsible mw-collapsed">seven proportional numbers<div class="mw-collapsible-content">
+{{#annotask:
+	[[category: #find a number]]
+	[[comment: seven proportional numbers]]
+}}</div></div><br>

Category	Comment	Link	Annotated text
word problem/find a number	perfect number	ספר_דיני_ממונות#x3Is	109) If you want to find the second [perfect number] קט)‫MS L: om. ואם תרצה למצא השני
word problem/find a number	perfect number	ספר_דיני_ממונות#IXt8	108) Question: if you want to find a perfect number קח) שאלה אם תרצה למצא {{#annot:term\|75\|xt6c}}מספר שלם{{#annotend:xt6c}} נקרא בלשונם {{#annot:term\|75\|aiWm}}נומירו פירפיטו{{#annotend:aiWm}}
word problem/find a number	perfect number	ספר_דיני_ממונות#CaJp	110) If you want to find the third [perfect number] קי)‫MS L: om. ואם ואם תרצה למצא השלישי

Category	Comment	Link	Annotated text
word problem/find a number	³/₇a+⁴/₅a=⅔a+¼a+20	ספר_מעשה_חושב#7YKC	Example: three-sevenths and four-fifths of the unknown number exceed two-thirds and a quarter of the unknown number by twenty and we wish to know how much is the number? :: $\scriptstyle\frac{3}{7}a+\frac{4}{5}a=\frac{2}{3}a+\frac{1}{4}a+20$ דמיון זה שיהיו ג' שביעיות וד' חמישיות המספר הנעלם מוסיפים על ב' שלישיות ורביעית המספר הנעלם עשרים ורצינו לדעת כמה המספר
word problem/find a number	³/₇a+⁴/₅a=⅔a+¼a+20	ספר_דיני_ממונות#SkQW	46) Question: three sevenths and four fifths of the unknown number exceed two thirds and a quarter of the unknown number by twenty. :We wish to know: how much is the number? : $\scriptstyle\frac{3}{7}a+\frac{4}{5}a=\frac{2}{3}a+\frac{1}{4}a+20$ מו) שאלה ג' שביעיות וד' חמישיות המספר הנעלם מוסיפים על ב' שלישיות ורביעית המספר הנעלם כ‫' רצינו לדעת כמה המספר הנעלם

Category	Comment	Link	Annotated text
word problem/find a number	repeated subtraction	מלאכת_המספר#TEa8	Question: if we subtract 12 from the double of any number, then we subtract another 12 from double the remainder also, and double the remainder is 12, how much is the original number? : $\scriptstyle2\sdot\left[\left[2\sdot\left(2X-12\right)\right]-12\right]=12$ שאילה אם מכפלו של אי זה מספר נקח י"ב ומכפל הנשאר נקח ג"כ י"ב אחרי' וכפל הנשאר יהיה י"ב כמה היה המספר הראשון
word problem/find a number	repeated subtraction	ספר_חשבון#QPoG	If one asks you: a number, you subtract a tenth from it, then you subtract again a tenth from the remainder, and 100 are left. :How much was the original number before subtracting the two tenths? : $\scriptstyle X-\frac{1}{10}X-\left[\frac{1}{10}\sdot\left(X-\frac{1}{10}X\right)\right]=100$ אם ישאלך אדם חשבון שהסרות ממנו הי' ומן הנשאר הסרות ממנו העשירית פעם שנית ונשארו ק‫' כמה היה כל החשבון בטרם שהסרת ממנו ב' העשיריות
word problem/find a number	repeated subtraction	מלאכת_המספר#j27H	Question: if the half of any amount plus one are subtracted, then the half of the remainder plus one, then the half of the remainder plus one, and one remains, how much is the original number? שאילה אם מאי זה כמות ילקח חציו ואחד יותר ומהנשאר חציו ויותר אחד ומהנשאר חציו ויותר אחד ונשאר אחד כמה היה המספר הראשון \|} : $\scriptstyle\left[\left[X-\left(\frac{1}{2}X+1\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+1\right)\right]\right]+1\right]\right]-\left[\left[\frac{1}{2}\sdot\left[\left[X-\left(\frac{1}{2}X+1\right)\right]-\left[\left[\frac{1}{2}\sdot\left[X-\left(\frac{1}{2}X+1\right)\right]\right]+1\right]\right]\right]+1\right]=1$

Category	Comment	Link	Annotated text
word problem/find a number	(a+b)\c=3⅖⅐;(a+c)\b=7⅔¼;b=30	ספר_דיני_ממונות#OJ9R	58) Question: when one number is added to a second number, their ratio to the third is 3 integers, 2-fifths and a seventh. :When the first is added to the third, their ratio to the [second] is 7 integers, 2-thirds and a quarter. :The second number is 30. :We wish to know: how much is each of the rest? נח) שאלה שיהיה מספר ראשון כשחובר אל מספר שני היה יחסו אל הג' ג' שלמים וב' חמישיות ושביעיות וכשחובר הראשון אל הג' היה יחסו הראשון ז' שלמים וב' שלישיות ורביעיות והמספר הב' ל‫' רצינו לדעת כמה כל א' מהנשארים
word problem/find a number	(a+b)\c=3⅖⅐;(a+c)\b=7⅔¼;b=30	ספר_מעשה_חושב#odgb	Example: when one number is added to a second number, their ratio to the third is 3 integers, 2-fifths and a seventh. ::When the first is added to the third, their ratio to the second is 7 integers, 2-thirds and a quarter. ::The second number is 30. ::We wish to know: how much is each of the rest? :: $\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=3+\frac{2}{5}+\frac{1}{7}\\\scriptstyle\left(a+c\right):b=7+\frac{2}{3}+\frac{1}{4}\\\scriptstyle b=30\end{cases}$ דמיון שיהיה מספר ראשון כשחובר אל מספר שני היה יחסו אל השלישי ג' שלמים וב' חמשיות ושביעית וכשחובר הראשון אל השלישי יהיה יחסו אל השני‫אל השני: P2271 הראשון ז' שלמים וב' שלישיות ורביעית והמספר השני שלשים ורצינו לדעת כמה כל אחד מהנשארים

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word problem/find a number	(a+b)/c=⅗+⅙;(a+c)/b=2⅓;a=20	ספר_דיני_ממונות#gfLj	59) Question: the ratio of the first and the second summed together to the third is 3-fifths plus a sixth. :The ratio of the first and the third summed together to the second is 2 integers and a third. :The first number is 20. :We wish to know: how much is each of the remaining numbers? נט) שאלה יהיה יחס הראשון עם השני מקובצים אל השלישי ג' חמישיות ושישית ויחס הראשון והג' מקובצים אל השני ב' שלמים ושלישית והמספר הראשון עשרים נרצה לדעת כמה כל א' מהמספרים הנשארים
word problem/find a number	(a+b)/c=⅗+⅙;(a+c)/b=2⅓;a=20	ספר_מעשה_חושב#0XvG	Another example: the ratio of the first and the second summed together to the third is 3-fifths and a sixth. ::The ratio of the first and the third summed together to the second is 2 integers and a third. ::The first number is 20. ::We wish to know: how much is each of the remaining numbers? :: $\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):c=\frac{3}{5}+\frac{1}{6}\\\scriptstyle\left(a+c\right):b=2+\frac{1}{3}\\\scriptstyle a=20\end{cases}$ דמיון אחר יהיה יחס הראשון והשני מקובצים אל השלישי ג' חמשיות וששית ויחס הראשון והשלישי מקובצים אל השני ב' שלמים ושלישית והמספר הראשון עשרים ורצינו לדעת כמה כל אחד מ[ה]מספרים הנשארים

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word problem/find a number	b/(c-a)=3⅓;c/(b-a)=6	ספר_מעשה_חושב#353o	Example: the ratio of the second to what remains from the third when the first is subtracted from it is 3 integers and a third. ::The ratio of the third to what remains from the second when the first is subtracted from it is the number 6. :: $\scriptstyle\begin{cases}\scriptstyle b:\left(c-a\right)=3+\frac{1}{3}\\\scriptstyle c:\left(b-a\right)=6\end{cases}$ והמשל שיהיה יחס ‫82vהשני אל הנשאר מהשלישי כשחוסר ממנו הראשון ג' שלמים ושליש ויחס השלישי אל הנשאר מהשני כשחוסר ממנו הראשון מספר ו'
word problem/find a number	b/(c-a)=3⅓;c/(b-a)=6	ספר_דיני_ממונות#73tk	60) Question: the ratio of the second to what remains from the third when the first is subtracted from it is 3 integers and a third. :The ratio of the third to what remains from the second when the first is subtracted from it is the number 6. ‫[ס)] שאלה שיהיה יחס הב' אל הנשאר מהג' כשחוסר ממנו הא' ג' שלמים ושליש ויחס הג' אל הנשאר מהב' כשחוסר ממנו הא' מספר ו‫'

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word problem/find a number	(a+b)/(c-a)=4½;(a+c)/(b-a)=5	ספר_מעשה_חושב#UAnP	Example: the ratio of the first and the second summed together to what remains from the third when the first is subtracted from it is 4 integers and a half. ::The ratio of the first and the third summed together to what remains from the second when the first is subtracted from it is 5 integers. ::We wish to know: how much is each of the numbers that meet this condition? :: $\scriptstyle\begin{cases}\scriptstyle\left(a+b\right):\left(c-a\right)=4+\frac{1}{2}\\\scriptstyle\left(a+c\right):\left(b-a\right)=5\end{cases}$ והמשל שיהיה יחס הראשון והשני מקובצים אל הנשאר [מהשלישי כשחוסר ממנו הראשון ד' שלמים וחצי ויחס הראשון והשלישי מקובצים אל הנשאר] מהשני כשחוסר ממנו הראשון הוא ה' שלמים ונרצה לדעת כמה כל המספרים שינהגו זה המנהג
word problem/find a number	(a+b)/(c-a)=4½;(a+c)/(b-a)=5	ספר_דיני_ממונות#UcvZ	61) Question: the ratio of the first and the second summed together to what remains from the third when the first is subtracted from it is 4 integers and a half. :The ratio of the first and the third summed together to what remains from the second when the first is subtracted from it is 5 integers. :We wish to know: how much is each of the numbers that meet this condition? סא) שאלה שיהיה יחס הראשון והשני מקובצים אל הנשאר מהג' כשחוסר ממנו הראשון ד' שלמים וחצי ויחס הראשון והג' מקובצים אל הנשאר מהב' כשחוסר ממנו הראשון הוא ה' שלמים ונרצה לדעת כמה כל המספרים שינהגו זה המנהג

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word problem/find a number	four proportional numbers	ספר_דיני_ממונות#Z5m3	Example: the first is 2 and the fourth is 16. דמיון המספר הראשון ב' והד' י"ו
word problem/find a number	four proportional numbers	ספר_דיני_ממונות#VcKa	Example: the first is 3-quarters and the last is 6 integers. דמיון הראשון ג' רביעיות והאחרון ו' שלמים

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word problem/find a number	five proportional numbers	ספר_דיני_ממונות#lGXB	257) Example of five numbers: the first is 3 and the last is 48. רנז) דמיון מה' מספרים הראשון ג' והאחרון מ"ח
word problem/find a number	five proportional numbers	ספר_דיני_ממונות#BxZ9	[2]52) Question: if you have 5 proportional numbers, the first is 3 and the fifth is 768, what is the second? קנב) שאלה אם יש לך ה' מספרים מתיחסים הא' ג' והה' תשס"ח מהו השני

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word problem/find a number	seven proportional numbers	ספר_דיני_ממונות#nCKV	258) Question of seven numbers: the first is 1 and the last is 729. רנח) שאלה מז' מספרים הראשון א' והאחרון תשכ"ט
word problem/find a number	seven proportional numbers	ספר_דיני_ממונות#By9s	259) Question of seven numbers with fractions: the first is an eighth and the seventh is 91 integers and an eighth. רנט) שאלה משברים מז' מספרים הראשון א' שמינית והז' צ"א שלמים ושמינית