Proportions appear in many forms in the Quantitative Reasoning section of the GMAT, and they tend to give test-takers trouble. A proportions problem may involve a basic ratio (for example, apples to bananas), varying rates (price per gallon), or geometric shapes (similar triangles). Recognizing proportion patterns and understanding the relationships they represent will give you an edge on Test Day.
Quantitative Reasoning problems involving ratios
A ratio is the relationship of one quantity to another, expressed in lowest terms. Think of a ratio as a reduced fraction of the relationship. Ratios may be expressed in words, such as, “The ratio of apples to bananas is three to four.” You may also see this ratio represented by a colon, in which case the question may say, “The ratio of apples to bananas is 3:4.” These are both legitimate, but they are mathematically pretty useless. In order to solve a problem involving a ratio, you need to express it written as a fraction.
The ratio of apples to bananas in a basket is 3:4. There are 12 apples in the basket. How many bananas are in the basket? First, write the ratio as a fraction:
Then plug in 12 for the total number of apples.
All you need to do now is cross-multiply and divide to solve for b, the number of bananas: 3b = 48, so b = 16.
If you know a:b and b:c, you can find a:c. Imagine that the ratio of roses to carnations in a flower shop is 2:5, and the ratio of carnations to tulips is 7:3. You could write that as follows:
We can’t just “smush” these ratios together to say roses:tulips = 2:3; we need to make the shared quantity the same. Both ratios include a value for carnations, but they are different values. Find the least common multiple of the different values to make them the same. Multiply each ratio as needed to combine:
If we are asked to find the ratio of roses to tulips at this flower shop, we ignore the number of carnations and only look at roses and tulips; the ratio is 14:15.
Proportions on the GMAT become a bit more tricky when they involve rates, but the same principles apply. Let’s say the price is $3 per gallon. If a customer purchases 12 gallons of gasoline, how much does she spend? Again, use a fraction to set up the proportion:
For rates, you can arrange the fractions in a couple of ways. You can pair price per gallon in one fraction, as I did here, or you could put the number of gallons together and the prices together:
Either way, when you cross-multiply you get x = 36, so the price of 12 gallons is $36.
Proportions commonly appear in GMAT geometry problems. Triangles involve several types of proportional relationships.
Triangle sides and angles have a proportional relationship. The side opposite the smallest angle in a triangle always has the shortest length, and the largest angle is always opposite the longest side. This is easy to remember with right triangles, as the hypotenuse is the longest side and it is opposite the 90°angle. If two angles are the same, the sides opposite them have the same length. This proportional relationship applies to all triangles and never changes.
Similar triangles have proportional sides and angles; their three angles have the same measures, and their side lengths share the same ratio.
In this figure, because lines AB and DE are parallel, triangles ABC and CDE are similar. Thus you know that the unmarked interior angles of triangle CDE are also 75° and 55°, respectively, and their side lengths are in proportion to one another.
“Pythagorean triplets” are right triangles with side lengths in tidy integer ratios. The most common triplets on the GMAT are 3:4:5 and 5:12:13. A 3:4:5 right triangle might have side lengths of 6:8:10 or 15:20:25; because the ratio is represented in the lowest terms; imagine an invisible x beside each term in the ratio. In a 6:8:10 triangle, for example, the x would be 2: 3(2):4(2):5(2).
Right triangles with angle measures of 30:60:90 and 45:45:90 are known as special right triangles and have side lengths in particular ratios as well. The side lengths of a 45:45:90 triangle always have a ratio of A 30:60:90 triangle always has side lengths of . Memorize these before test day to make quick work of solving problems that use special right triangles.
Brushing up on proportions is an important part of preparing for GMAT Quantitative Reasoning. Look for proportional relationships in problems that don’t look like proportion problems; recognize this pattern on test day.