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Why when you have to use the compound rule of three that has inverse proportions do you need to invert a fraction and multiply them?

Example: A daycare center with 250 children provides 4 portions of food per day to each child for 18 days. If the population increases to 50 children, how many days will the food last if it is reduced to 3 daily servings?

Solution:

  • Ratios are formed between the quantities.
  • The more children, the food lasts fewer days, therefore the proportion is inverse.
  • The fewer portions the food lasts for more days, therefore the proportion is inverse.

250 kids // 4 portions // 18 days

300 kids // 3 portions // x days

The ratios 250/300 and 4/3 are inverted and multiplied, the ratio 18/x is equal to the product.

(300/250)(3/4) = 18/x

x = 20

Therefore, the food will last 20 days.

Alejandra
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1 Answers1

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I like to consider what quantity is kept constant; here it's the total inventory of food:

$$\begin{align*} 250\text{ children} \cdot 4\text{ portions of food $/$ day $/$ child}\cdot 18\text{ days} &= I \text{ portions of food}\tag {1}\\ 300\text{ children} \cdot 3\text{ portions of food $/$ day $/$ child}\cdot x\text{ days} &= I \text{ portions of food}\tag {2} \end{align*}$$

Note how the units on both sides match; the units also hint at whether one should multiply or divide quantities.

One may first use $(1)$ to find $I$, then use $(2)$ to find $x$. Or to follow the question, divide $(2)/(1)$,

$$\begin{align*} \frac{300}{250} \cdot \frac34 \cdot \frac x{18} &= 1\\ \frac{300}{250} \cdot \frac34&= \frac{18}x\\ x &= 20\\ x\text{ days} &= 20\text{ days} \end{align*}$$

peterwhy
  • 22,256