# Proportion Word Problems

- - Pre-Algebra

After learning proportions, let us now try to solve word problems involving proportions. To do so, we form the proportion between the ratios given in the problem then solve for the missing value by applying the property between its means and extremes.

Consider the examples below:

Example 1: John Dale spent P180 for 12 meters of rope. How much will 8 meters of rope cost? If John Dale spent P180 for 12 meters of rope, will the cost of 8 meters of rope be greater than or lesser than P180?

Solution:

If you decrease the number of meters of rope, there is a consequent decrease in the cost. This proportion is called direct proportion. To solve for the missing quantity, express the two values of the same quantity (unit) as one ratio and their corresponding pair of quantity as the second ratio.

Let $x$ be the cost of 8 meters of rope:

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle \frac{12}{8}&=&\displaystyle \frac{180}{x} \\ &12x&=&8\left (180 \right ) \\ &12x&=& 1440 \\ \therefore &x&=&120 \\ \end{array}$

Therefore, the cost of 8 meters of rope is P120.

Example 2: If it takes Myrna 30 minutes to walk 3–km distance, at the same rate, how long will it take her to walk a 10–km distance?

Solution:

At the same rate, it would take Myrna a longer time to travel a longer distance. Again, express the two values of the same quantity (unit) as one ratio and their corresponding pair of quantity as the second ratio.

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle \frac{30}{x}&=&\displaystyle \frac{3}{10} \\ &3x&=&300 \\ \therefore &x&=&100 \\ \end{array}$

Therefore, it will take Myrna 100 minutes or 1 hour and 40 minutes to walk a 10–km distance.

Example 3: If it takes 35 meters of cloth to make 3 rectangular tablecloths of the same area, how many meters of cloth are needed to make 5 rectangular tablecloths of the same dimensions?

Solution:

If you increase the number of tablecloths to make, there corresponds an increase in the number of meters of cloth required.

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle \frac{35}{x}&=&\displaystyle \frac{3}{5} \\ &3x&=&175 \\ \therefore &x&\approx &100 \\ \end{array}$

Approximately, $58 \frac {1}{3}$ meters of cloth are needed to make 5 rectangular tablecloths.

Example 4: If it takes 5 men to finish a painting job in 8 days, how many men are required to finish the same job in 4 days, considering that all men will work at the same pace?

Solution:

If you want to reduce the number of days to finish a certain job, there must be a consequent increase in the number of people to work. This is an example of an indirect proportion.

To form the proportion, place the bigger number in each ratio on the same side, either as extremes or as means. The proportion is

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle \frac{5}{4}&=&\displaystyle \frac{x}{8} \\ &4x&=&40 \\ \therefore &x&=&10 \\ \end{array}$

Therefore, it needs 10 men to finish the painting job in 4 days.