Fractions, ratios, and decimals: complete GMAT guide

A fraction $a/b$ compares a part $a$ to a whole $b$. The whole must be defined. If a class has 12 boys and 18 girls, the fraction of boys is $12/30=2/5$, not $12/18$. The ratio of boys to girls is $12:18=2:3$, which compares parts, not a part to the whole.

A ratio compares two or more quantities. The ratio $3:4$ means that for every 3 units of the first quantity there are 4 units of the second. The actual quantities can be $3k$ and $4k$ for some positive number $k$. This multiplier is the key to ratio questions.

A proportion sets two ratios equal: $\frac{a}{b}=\frac{c}{d}$. Cross-multiply to get $ad=bc$. Keep the units in the same order on both sides. If the left ratio is miles per hour, the right ratio must also be miles per hour.

To combine $a:b$ and $b:c$, make the shared term the same in both ratios. If the ratio of men to women is 2:3 and the ratio of women to children is 4:5, rewrite the first ratio with 12 women: $8:12$. Rewrite the second with 12 women: $12:15$. The combined ratio is $8:12:15$.

If the shared term is not the same, multiply each ratio by the factor that makes the shared term equal. This is the same as finding a common denominator for the shared quantity.

If a mixture has parts in the ratio $2:3:5$, the whole has $2+3+5=10$ parts. The first part is $2/10=1/5$ of the whole, the second is $3/10$, and the third is $5/10=1/2$. To convert a part-to-part ratio into a part-to-whole fraction, add the parts to get the whole.

When a question says "one third of the students who play chess also play Go," multiply the fractions. If $3/4$ of a club plays chess and $1/3$ of those play Go, the fraction who play both is $(3/4)(1/3)=1/4$. The word "of" usually signals multiplication.

For successive changes, multiply the multipliers. A price increases by 20% and then decreases by 20%. The final price is $1.2 \times 0.8=0.96$ times the original, a 4% decrease. The two percentages do not cancel because they apply to different bases.

Each place to the right of the decimal point represents a negative power of ten. In 0.472, the 4 is tenths, the 7 is hundredths, and the 2 is thousandths. To convert a decimal to a fraction, write the digits over the appropriate power of ten and simplify. $0.472=472/1000=59/125$.

To compare decimals, line up the decimal points and compare digit by digit. Do not compare by the number of digits after the decimal. $0.7$ is greater than $0.65$ because 0.70 is greater than 0.65.

A percent is a ratio out of 100. $35\%=35/100=0.35$. To find a percent of a number, multiply by the decimal form. To find what percent one number is of another, divide the part by the whole and multiply by 100.

A rate compares quantities with different units. A unit rate expresses the comparison per one unit of the second quantity. If 5 pounds of apples cost $12.50, the unit price is $12.50/5=2.50$ dollars per pound. To compare two products, convert both to the same unit rate.

A scale is a ratio between a drawing and the actual object. If 1 inch represents 20 feet, then 3.5 inches represent 70 feet. Keep the drawing units on one side and the actual units on the other.

Some ratio questions change the total by adding or removing items. If the ratio of boys to girls is 3:4 and 6 boys join the class, the new ratio may be given. To solve, write the original numbers as $3k$ and $4k$, add the change to the appropriate part, and set up the new ratio. Do not apply the change to both parts unless the problem says so.

When a problem gives a fraction of a group and then a fraction of the remainder, work step by step. If $1/3$ of a class plays chess and $1/2$ of the remaining students play Go, the fraction who play neither is $(2/3)(1/2)=1/3$. The fractions apply to different groups, so you cannot simply add or subtract them without tracking the base.

A common error is to subtract the first fraction from 1 and then apply the second fraction to the original total. Apply each fraction to the group that remains after the previous step.

1. Identify the whole and the parts. Decide whether the question is part-to-part or part-to-whole.
2. Write ratios with a multiplier $k$ when actual quantities are involved.
3. Combine ratios by making the shared term equal.
4. Convert among fractions, decimals, and percents as needed.
5. For successive changes, multiply the multipliers rather than adding percentages.
6. Check that the answer has the requested form.

• Using the wrong denominator when finding a fraction of a group.
• Adding percentages that apply to different bases.
• Combining ratios without making the shared term equal.
• Treating a part-to-part ratio as a part-to-whole fraction.
• Comparing decimals by the number of digits.
• Forgetting to convert a percent to a decimal before multiplying.
• Reversing the order of a rate or scale.