# Problem of the Week

## Updated at Jun 24, 2019 1:16 PM

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

How would you solve the equation $$(5+\frac{5}{w})\times \frac{2}{2+w}=\frac{12}{7}$$?

Here are the steps:

$(5+\frac{5}{w})\times \frac{2}{2+w}=\frac{12}{7}$

 1 Expand.$\frac{10}{2+w}+\frac{10}{w(2+w)}=\frac{12}{7}$2 Multiply both sides by the Least Common Denominator: $$7w(2+w)$$.$70w+70=12w(2+w)$3 Simplify.$70w+70=24w+12{w}^{2}$4 Move all terms to one side.$70w+70-24w-12{w}^{2}=0$5 Simplify  $$70w+70-24w-12{w}^{2}$$  to  $$46w+70-12{w}^{2}$$.$46w+70-12{w}^{2}=0$6 Factor out the common term $$2$$.$2(23w+35-6{w}^{2})=0$7 Factor out the negative sign.$2\times -(6{w}^{2}-23w-35)=0$8 Divide both sides by $$2$$.$-6{w}^{2}+23w+35=0$9 Multiply both sides by $$-1$$.$6{w}^{2}-23w-35=0$10 Split the second term in $$6{w}^{2}-23w-35$$ into two terms.1 Multiply the coefficient of the first term by the constant term.$6\times -35=-210$2 Ask: Which two numbers add up to $$-23$$ and multiply to $$-210$$?$$7$$ and $$-30$$3 Split $$-23w$$ as the sum of $$7w$$ and $$-30w$$.$6{w}^{2}+7w-30w-35$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$6{w}^{2}+7w-30w-35=0$11 Factor out common terms in the first two terms, then in the last two terms.$w(6w+7)-5(6w+7)=0$12 Factor out the common term $$6w+7$$.$(6w+7)(w-5)=0$13 Solve for $$w$$.1 Ask: When will $$(6w+7)(w-5)$$ equal zero?When $$6w+7=0$$ or $$w-5=0$$2 Solve each of the 2 equations above.$w=-\frac{7}{6},5$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$w=-\frac{7}{6},5$Done Decimal Form: -1.166667, 5w=-7/6,5