These questions move between ratios, fractions of a whole and comparisons between parts. The decisive step is to identify what the denominator represents before calculating.
Name the numerator and denominator
Treat the terms of a ratio as counts of equal parts. In a two-part ratio a:b, the combined whole contains a+b parts. The first share is therefore a/(a+b), and the second share is b/(a+b). For example, a 4:6 ratio contains 10 parts, so the first share is 4/10, which simplifies to 2/5.
| Question form | Method |
|---|---|
| Fraction of the whole | Use the requested part as numerator and the sum of every ratio term as denominator, then simplify. |
| One part compared with another | Use the two relevant ratio terms in the named order; do not replace the second term with the total. |
| A fraction of the whole is given and the rest forms one group | Keep the stated numerator as that group's parts and subtract it from the denominator to find the remaining group's parts. |
| Three or more groups | Add every term for the whole, even when only one group's fraction is requested. |
For a reverse conversion, suppose one group is 7/12 of a collection and all remaining objects form the other group. The remaining share is 5/12, so the part-to-part ratio is 7:5. The denominator counts the whole; it does not become the second group's term.
Where mistakes happen
| Tempting route | How to reject it |
|---|---|
| Put the other part directly under the requested part | That gives a part-to-part quotient. If the wording says of all or of the whole, add all ratio terms for the denominator. |
| Use the other group's term as the numerator | Match the requested group to its own position in the ratio before building the fraction. |
| Turn a fraction of the whole straight into a ratio against its denominator | The denominator includes both groups. Subtract the known numerator from it to find the complementary group first. |
| Ignore one term of a three-part ratio | A fraction of the whole needs every group's parts in the denominator, including groups not named in the numerator. |
| Treat a quotient greater than one as a share of a positive whole | A proper part of such a whole cannot exceed the whole. A value above one signals that you formed a comparison between parts instead. |
Try it
Blue and red flags are in the ratio 3:5. The fraction of all flags that are blue is 3/d. Enter the denominator d.
The ratio of boys to girls in a club is 4:9. What fraction of the club members are girls?
- 4/13
- 4/9
- 9/4
- 9/13
Quantities p and q are in the ratio 5:2. Which statements correctly distinguish a ratio from a fraction of the whole? Select all that apply.
- p is 5/7 of the whole.
- q is 2/7 of the whole.
- p is 5/2 times q.
- p is 5/2 of the whole.
- q is 5/7 times p.