Problem of the Week

Updated at Oct 16, 2023 12:46 PM

For this week we've brought you this equation problem.

How would you solve the equation (p−35)26=275\frac{{(\frac{p-3}{5})}^{2}}{6}=\frac{2}{75}?

Here are the steps:



(p−35)26=275\frac{{(\frac{p-3}{5})}^{2}}{6}=\frac{2}{75}

1
Use Division Distributive Property: (xy)a=xaya{(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}.
(p−3)2526=275\frac{\frac{{(p-3)}^{2}}{{5}^{2}}}{6}=\frac{2}{75}

2
Simplify  52{5}^{2}  to  2525.
(p−3)2256=275\frac{\frac{{(p-3)}^{2}}{25}}{6}=\frac{2}{75}

3
Simplify  (p−3)2256\frac{\frac{{(p-3)}^{2}}{25}}{6}  to  (p−3)225×6\frac{{(p-3)}^{2}}{25\times 6}.
(p−3)225×6=275\frac{{(p-3)}^{2}}{25\times 6}=\frac{2}{75}

4
Simplify  25×625\times 6  to  150150.
(p−3)2150=275\frac{{(p-3)}^{2}}{150}=\frac{2}{75}

5
Multiply both sides by 150150.
(p−3)2=275×150{(p-3)}^{2}=\frac{2}{75}\times 150

6
Use this rule: ab×c=acb\frac{a}{b} \times c=\frac{ac}{b}.
(p−3)2=2×15075{(p-3)}^{2}=\frac{2\times 150}{75}

7
Simplify  2×1502\times 150  to  300300.
(p−3)2=30075{(p-3)}^{2}=\frac{300}{75}

8
Simplify  30075\frac{300}{75}  to  44.
(p−3)2=4{(p-3)}^{2}=4

9
Take the square root of both sides.
p−3=±4p-3=\pm \sqrt{4}

10
Since 2×2=42\times 2=4, the square root of 44 is 22.
p−3=±2p-3=\pm 2

11
Break down the problem into these 2 equations.
p−3=2p-3=2
p−3=−2p-3=-2

12
Solve the 1st equation: p−3=2p-3=2.
p=5p=5

13
Solve the 2nd equation: p−3=−2p-3=-2.
p=1p=1

14
Collect all solutions.
p=5,1p=5,1

Done
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