Toothpaste Cost Comparison
A consumer report revealed the following information about three tubes of toothpaste. Bright is $ 60 \% $ more expensive than Fresh and has $ 25\% $ less volume than Glow. Glow is $ 25\% $ less expensive than Bright and has $ 33\frac{1}{3} \% $ more volume than Fresh. Fresh costs $ \$1.00$ per unit of volume. What is the number of cents per unit of volume of Glow?
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Solution
Make a table to organize the information about the cost and the volume of the three tubes of toothpaste. Let the number of units of volume in one tube of Fresh be $ u $. Then the cost of one tube of Fresh is $ \$u$ . Since Bright is more expensive than Fresh by $60\%$ , one tube of Bright costs $\$ \,\frac{8}{5}u $. Also, since Glow has $ 33\frac{1}{3}\% $ more volume than Fresh, the volume of a tube of Glow is $ \frac{4}{3}u $ units. \[
\begin{array}{c|cc}
& \text{volume} & \text{cost} \\ \hline
\text{Bright} & & \$ \,\frac{8}{5}u \\
\text{Fresh} & u & \$\,u\\
\text{Glow} & \frac{4}{3}u &
\end{array}
\]Finally, multiply the volume of Glow by $ \frac{3}{4} $ to find the volume of Bright, and multiply the cost of Bright by $ \frac{3}{4} $ to find the cost of Glow. \[
\begin{array}{c|cc}
& \text{volume} & \text{cost} \\ \hline
\text{Bright} & u & \$ \,\frac{8}{5}u \\
\text{Fresh} & u & \$\,u\\
\text{Glow} & \frac{4}{3}u & \$ \,\frac{6}{5} u
\end{array}
\]Dividing the cost of a tube of Glow by its volume, we find a cost-per-unit-volume of $\$ \frac{6}{5}u\div \frac{4}{3}u=\$\frac{9}{10}=\boxed{90}$ cents.
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