# Problem of the Week

## Updated at Jul 19, 2021 4:12 PM

How can we solve the equation $${(\frac{5}{n})}^{2}\times \frac{3-n}{2}=25$$?

Below is the solution.

${(\frac{5}{n})}^{2}\times \frac{3-n}{2}=25$

 1 Use Division Distributive Property: $${(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}$$.$\frac{{5}^{2}}{{n}^{2}}\times \frac{3-n}{2}=25$2 Simplify  $${5}^{2}$$  to  $$25$$.$\frac{25}{{n}^{2}}\times \frac{3-n}{2}=25$3 Use this rule: $$\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}$$.$\frac{25(3-n)}{{n}^{2}\times 2}=25$4 Regroup terms.$\frac{25(3-n)}{2{n}^{2}}=25$5 Multiply both sides by $$2{n}^{2}$$.$25(3-n)=25\times 2{n}^{2}$6 Simplify  $$25\times 2{n}^{2}$$  to  $$50{n}^{2}$$.$25(3-n)=50{n}^{2}$7 Divide both sides by $$25$$.$3-n=2{n}^{2}$8 Move all terms to one side.$3-n-2{n}^{2}=0$9 Multiply both sides by $$-1$$.$2{n}^{2}+n-3=0$10 Split the second term in $$2{n}^{2}+n-3$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$2\times -3=-6$2 Ask: Which two numbers add up to $$1$$ and multiply to $$-6$$?$$3$$ and $$-2$$3 Split $$n$$ as the sum of $$3n$$ and $$-2n$$.$2{n}^{2}+3n-2n-3$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$2{n}^{2}+3n-2n-3=0$11 Factor out common terms in the first two terms, then in the last two terms.$n(2n+3)-(2n+3)=0$12 Factor out the common term $$2n+3$$.$(2n+3)(n-1)=0$13 Solve for $$n$$.1 Ask: When will $$(2n+3)(n-1)$$ equal zero?When $$2n+3=0$$ or $$n-1=0$$2 Solve each of the 2 equations above.$n=-\frac{3}{2},1$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$n=-\frac{3}{2},1$Done Decimal Form: -1.5, 1n=-3/2,1