Word problem about percentages. Represent as an equation and explain

Question

A vendor sells only Product A, for \$6, and Product B, for \$21. If Q% of the products sold are Product B, and if T% of the total revenue comes from sales of Product B, find Q in terms of T.

I can not get the algebraic equation right.

Answer 1

Let $a$ be the quantity of Product A sold. Let $b$ be the quantity of Product B sold. Then the fraction of products sold that are Product A is $$ \frac{a}{a+b} $$ and the fraction of products sold that are Product B is $$ \frac{b}{a+b} \text{.} $$ (Note that these sum to $\frac{a}{a+b} + \frac{b}{a+b} = \frac{a+b}{a+b} = 1$, so we have accounted for all the products sold.) To convert either of these fractions to a percentage, multiply by $100\%$.

Revenue from Product A is $\$6 a$ and revenue from Product B is $\$21 b$, so total revenue is $\$6 a + \$21 b$. The fractions of revenue from Product A and Product B are, respectively, $$ \frac{\$6 a}{\$6 a + \$21 b} \text{ and } \frac{\$21 b}{\$6 a + \$21 b} \text{.} $$ As before, we can convert either to percentage by multiplying by $100\%$.

We are told that $$ Q\% = \frac{b}{a+b} \times 100\% $$ and $$ T\% = \frac{\$21 b}{\$6 a + \$21 b} \times 100\% \text{.} $$

Now we compute, first cancelling "$\%$" and "$\$$", \begin{align*} \frac{T}{100} &= \frac{21 b}{6 a + 21 b} \text{, so } \\ 1 - \frac{T}{100} &= \frac{6 a + 21 b}{6 a + 21 b} - \frac{21 b}{6 a + 21 b} \\ &= \frac{6 a}{6 a + 21 b} \text{.} \end{align*} Then, finding a combination that gives us $a+b$, \begin{align*} 6 \cdot \frac{T}{100} + 21 \cdot \left( 1 - \frac{T}{100} \right) &= \frac{6 \cdot 21 b}{6 a + 21 b} + \frac{21 \cdot 6 a}{6 a + 21 b} \\ &= \frac{6 \cdot 21 (a+b)}{6 a + 21 b} \text{.} \end{align*} We want that $a+b$ in the denominator so we take reciprocals and use $\frac{T}{100}$ to get the lone $b$ in the numerator. \begin{align*} \frac{1}{6 \cdot \frac{T}{100} + 21 \cdot \left( 1 - \frac{T}{100} \right)} &= \frac{6 a + 21 b}{6 \cdot 21 (a+b)} \text{, so} \\ 6 \cdot \frac{T}{100} \cdot \frac{1}{6 \cdot \frac{T}{100} + 21 \cdot \left( 1 - \frac{T}{100} \right)} &= 6 \cdot \frac{21 b}{6 a + 21 b} \cdot \frac{6 a + 21 b}{6 \cdot 21 (a+b)} \\ &= \frac{6 \cdot 21 b}{6 \cdot 21 (a+b)} \\ &= \frac{b}{a+b} \\ &= \frac{Q\%}{100\%} \text{.} \end{align*}

This gives \begin{align*} Q &= \frac{6T}{6 \cdot \frac{T}{100} + 21 \cdot \left( 1 - \frac{T}{100} \right)} \cdot \frac{100}{100} \\ &= \frac{600T}{6 T + 21 \cdot \left( 100 - T \right)} \\ &= \frac{600T}{6 T + 2100 - 21T} \\ &= \frac{600T}{2100 - 15T } \\ &= \frac{15 \cdot 40T}{15 \cdot (140 - T) } \\ &= \frac{40T}{140 - T} \text{.} \end{align*}