Two numbers x and y are such that the sum of 4% of x and 5% of y is two thirds of the sum of 5% of x and 8% of y. Find the ratio x : y.

Step-by-step solution

1. 1

   Turn the words into symbols. "$p\%$ of $x$" means $\tfrac{p}{100}x$, so the sentence becomes

   $\frac{4}{100}x+\frac{5}{100}y=\frac{2}{3}\left(\frac{5}{100}x+\frac{8}{100}y\right)$

   One equation in two unknowns cannot pin down $x$ and $y$ individually — but it can fix their ratio, which is exactly what is asked.

2. 2

   Clear the hundredths. Every term carries a factor $\tfrac{1}{100}$, so multiplying both sides by $100$ removes it entirely:

   $4x+5y=\frac{2}{3}(5x+8y)$

3. 3

   Clear the fraction. Multiply both sides by $3$:

   $3(4x+5y)=2(5x+8y)\quad\Longrightarrow\quad 12x+15y=10x+16y$

4. 4

   Collect the variables on opposite sides.

   $12x-10x=16y-15y\quad\Longrightarrow\quad 2x=y$

5. 5

   Convert to a ratio. From $2x=y$,

   $\frac{x}{y}=\frac{1}{2}\quad\Longrightarrow\quad x:y=1:2$

   $\boxed{1:2}$

6. 6

   Check with actual numbers. Take $x=1$, $y=2$: the left side is $0.04(1)+0.05(2)=0.14$, and the right side is $\tfrac23\big(0.05(1)+0.08(2)\big)=\tfrac23(0.21)=0.14$. Any pair in the ratio $1:2$, such as $x=50$, $y=100$, works equally well — the equation is homogeneous, so only the ratio matters.