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 32+52+u=911\frac{3}{2+\frac{5}{2+u}}=\frac{9}{11}?

Let's start!



32+52+u=911\frac{3}{2+\frac{5}{2+u}}=\frac{9}{11}

1
Multiply both sides by 2+52+u2+\frac{5}{2+u}.
3=911(2+52+u)3=\frac{9}{11}(2+\frac{5}{2+u})

2
Divide both sides by 99.
39=111(2+52+u)\frac{3}{9}=\frac{1}{11}(2+\frac{5}{2+u})

3
Simplify  39\frac{3}{9}  to  13\frac{1}{3}.
13=111(2+52+u)\frac{1}{3}=\frac{1}{11}(2+\frac{5}{2+u})

4
Simplify  2+52+u11\frac{2+\frac{5}{2+u}}{11}  to  211+52+u11\frac{2}{11}+\frac{\frac{5}{2+u}}{11}.
13=211+52+u11\frac{1}{3}=\frac{2}{11}+\frac{\frac{5}{2+u}}{11}

5
Simplify  52+u11\frac{\frac{5}{2+u}}{11}  to  511(2+u)\frac{5}{11(2+u)}.
13=211+511(2+u)\frac{1}{3}=\frac{2}{11}+\frac{5}{11(2+u)}

6
Subtract 211\frac{2}{11} from both sides.
13−211=511(2+u)\frac{1}{3}-\frac{2}{11}=\frac{5}{11(2+u)}

7
Simplify  13−211\frac{1}{3}-\frac{2}{11}  to  533\frac{5}{33}.
533=511(2+u)\frac{5}{33}=\frac{5}{11(2+u)}

8
Multiply both sides by 11(2+u)11(2+u).
533×11(2+u)=5\frac{5}{33}\times 11(2+u)=5

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

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

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

12
Multiply both sides by 33.
5(2+u)=5×35(2+u)=5\times 3

13
Simplify  5×35\times 3  to  1515.
5(2+u)=155(2+u)=15

14
Divide both sides by 55.
2+u=1552+u=\frac{15}{5}

15
Simplify  155\frac{15}{5}  to  33.
2+u=32+u=3

16
Subtract 22 from both sides.
u=3−2u=3-2

17
Simplify  3−23-2  to  11.
u=1u=1

Done
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