Proportions

If 3 notebooks cost 4.50 dollars, 7 cost 10.50, because 3/4.50 = 7/10.50.

Three notebooks for 4.50. What do seven cost?

Try this before reading on, without any formula. Three notebooks cost 4.50 dollars. What do seven cost? Write your answer down. One route is to find the price of one notebook first. Another is tempting and wrong: seven is four more than three, so add four dollars. The answer comes back further down, once the tool behind the first route has a name.

A proportion says two ratios are equal

OpenStax defines a proportion as an equation of the form a/b = c/d, where b and d are not zero. It says that two ratios or rates are equal, and it is read "a is to b as c is to d". The statement 2/3 = 8/12 is a proportion: 2 is to 3 as 8 is to 12. So is 3 notebooks / 4.50 dollars = 7 notebooks / 10.50 dollars.

The bars show 3/4 and 6/8 shaded to the same length. 3 : 4 = 6 : 8 is a proportion because both ratios describe the same amount. Proportions are equivalent fractions in another setting.

Outer times outer equals inner times inner

For any proportion a/b = c/d, the cross products are equal: a × d = b × c. Check 2/3 = 8/12: 2 × 12 = 24 and 3 × 8 = 24. They match, so the proportion is true. Check 3/5 and 4/7: 3 × 7 = 21 and 5 × 4 = 20. They differ, so 3/5 and 4/7 are not a proportion. In the colon form, the outer numbers multiplied equal the inner numbers multiplied.

Solving x/12 = 5/8, and a shortcut for 3/4 = x/20

Most proportion problems leave one number blank. Solve x/12 = 5/8. Cross-multiply: 8 × x = 12 × 5 = 60, so x = 60 ÷ 8 = 7.5. Check: 7.5/12 = 0.625 and 5/8 = 0.625. Sometimes there is a quicker way. In 3/4 = x/20, the denominator went from 4 to 20, which is × 5, so x = 3 × 5 = 15.

Back to the notebooks: 10.50 dollars

So, the notebooks. The main job in a word problem is setting up both ratios the same way round. Write notebooks over dollars on both sides: 3/4.50 = 7/x. Then 3x = 4.50 × 7 = 31.50, and x = 10.50 dollars. The adding shortcut gives 4.50 + 4 = 8.50, which is too little. Put notebooks on top on one side and dollars on top on the other and the arithmetic still runs, but the answer is nonsense.

30% of 70 as a proportion

OpenStax writes percent problems as a proportion too: the amount is to the base as the percent is to 100, or amount/base = percent/100. What is 30% of 70? x/70 = 30/100, so 100x = 2,100 and x = 21. 18 is what percent of 72? 18/72 = p/100, so 72p = 1,800 and p = 25.

A taxi fare that breaks the rule

A proportion only fits when doubling one quantity doubles the other. A made-up taxi charges a 4-dollar starting fee plus 2 dollars per kilometre. 5 km costs 4 + 10 = 14 dollars, but 10 km costs 4 + 20 = 24 dollars, not 28. The fixed fee breaks the proportion. Before setting up a proportion, ask whether zero of one thing would mean zero of the other.