Problem of the Week

Updated at Mar 10, 2025 12:48 PM

This week we have another equation problem:

How can we solve the equation \(\frac{3}{2+\frac{5}{2+u}}=\frac{9}{11}\)?

Let's start!



\[\frac{3}{2+\frac{5}{2+u}}=\frac{9}{11}\]

1
Multiply both sides by \(2+\frac{5}{2+u}\).
\[3=\frac{9}{11}(2+\frac{5}{2+u})\]

2
Divide both sides by \(9\).
\[\frac{3}{9}=\frac{1}{11}(2+\frac{5}{2+u})\]

3
Simplify  \(\frac{3}{9}\)  to  \(\frac{1}{3}\).
\[\frac{1}{3}=\frac{1}{11}(2+\frac{5}{2+u})\]

4
Simplify  \(\frac{2+\frac{5}{2+u}}{11}\)  to  \(\frac{2}{11}+\frac{\frac{5}{2+u}}{11}\).
\[\frac{1}{3}=\frac{2}{11}+\frac{\frac{5}{2+u}}{11}\]

5
Simplify  \(\frac{\frac{5}{2+u}}{11}\)  to  \(\frac{5}{11(2+u)}\).
\[\frac{1}{3}=\frac{2}{11}+\frac{5}{11(2+u)}\]

6
Subtract \(\frac{2}{11}\) from both sides.
\[\frac{1}{3}-\frac{2}{11}=\frac{5}{11(2+u)}\]

7
Simplify  \(\frac{1}{3}-\frac{2}{11}\)  to  \(\frac{5}{33}\).
\[\frac{5}{33}=\frac{5}{11(2+u)}\]

8
Multiply both sides by \(11(2+u)\).
\[\frac{5}{33}\times 11(2+u)=5\]

9
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{5\times 11(2+u)}{33}=5\]

10
Simplify  \(5\times 11(2+u)\)  to  \(55(2+u)\).
\[\frac{55(2+u)}{33}=5\]

11
Simplify  \(\frac{55(2+u)}{33}\)  to  \(\frac{5(2+u)}{3}\).
\[\frac{5(2+u)}{3}=5\]

12
Multiply both sides by \(3\).
\[5(2+u)=5\times 3\]

13
Simplify  \(5\times 3\)  to  \(15\).
\[5(2+u)=15\]

14
Divide both sides by \(5\).
\[2+u=\frac{15}{5}\]

15
Simplify  \(\frac{15}{5}\)  to  \(3\).
\[2+u=3\]

16
Subtract \(2\) from both sides.
\[u=3-2\]

17
Simplify  \(3-2\)  to  \(1\).
\[u=1\]

Done
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