These questions ask for one share, every share or a total when quantities are divided in a stated ratio. The same unitary method works whether you begin with the total, one share or the difference between two shares.
Find one ratio part first
When the total is known, add the ratio terms and divide the total by that sum. For example, sharing 96 in the ratio 3:5 gives 8 parts, so one part is 12; the shares are 36 and 60. Multiply the one-part value by each term in the original order, then check that the shares add to the total.
| What is known | Route to one part |
|---|---|
| The total | Divide the total by the sum of the ratio terms. |
| One share | Divide that share by its matching ratio term. For example, if a 6-part share is 42, one part is 7. |
| The difference between two shares | Subtract the relevant ratio terms, then divide the stated difference by that number of parts. For example, a 20-unit difference in a 4:6 ratio makes one part worth 10. |
| Three or more shares | Keep the named order, add every term for the total part count, and multiply one part by each term. |
After finding one part, build only what the question requests. A combined total uses the sum of all terms; a particular share uses its own term; a difference uses the difference between terms. Two quick checks catch many slips: the shares must have the stated order and ratio, and their sum must equal the known total.
Where mistakes happen
| Tempting route | How to reject it |
|---|---|
| Divide the total separately by each ratio term | The terms describe how many equal parts each person or group receives. Divide by the sum first, then multiply. |
| Choose shares that add correctly but do not have the stated ratio | A correct total is only one check. Divide the proposed shares by their corresponding terms; both results must give the same one-part value. |
| Treat a ratio term as an amount to add to a known share | A term is a multiplier, not a measurement or money amount. Divide the known share by its term to recover the common part value. |
| Reverse the two shares | Attach each term to the person or group named in the same position before multiplying. |
| Divide by only one term when the total is known | The total contains every ratio part, so its divisor is the sum of all terms rather than any single term. |
Try it
Two lengths are in the ratio 7:4 and differ by 18 cm. Enter their combined length in cm.
Seventy-two tokens are shared among Ari, Bea, and Chen in the ratio 1:2:3. Which sharing-in-a-ratio statements are true? Select all that apply.
- One ratio part is 12 tokens.
- Ari receives 12 tokens.
- Bea receives 24 tokens.
- Chen receives 36 tokens.
- Bea receives 18 tokens.
A team allocates 168 minutes to planning, building, and checking in the ratio 3:4:5. Which allocation is correct?
- 42, 60, and 66 minutes
- 56, 42, and 70 minutes
- 36, 48, and 84 minutes
- 42, 56, and 70 minutes