Count the parts first
Amy and Bo share 20 sweets in the ratio 2 : 3. The ratio says the sweets are split into 2 + 3 = 5 equal parts: Amy gets 2 of the parts and Bo gets 3 of them. We do not know yet how many sweets are in one part.
5 equal parts share the 20 sweets. What one part is worth is not known yet.
Find one part, then multiply
All 5 parts together hold 20 sweets, so one part is worth 20 ÷ 5 = 4 sweets. Once you know what one part is worth, every share is just a number of parts. Amy has 2 parts, which is 2 × 4 = 8 sweets. Bo has 3 parts, which is 3 × 4 = 12 sweets. Check by adding the shares: 8 + 12 = 20, the number of sweets we started with.
Finding the value of one part and then multiplying is called the unitary method. "Unit" means one.
20 ÷ 5 = 4 in every part. Amy's 2 parts are 8 sweets and Bo's 3 parts are 12.
When you know one share
Sometimes a question gives one share instead of the total. Amy and Bo share some sweets in the ratio 2 : 3, and Bo gets 15. Bo's 3 parts hold 15 sweets, so one part is 15 ÷ 3 = 5 sweets. Then Amy has 2 × 5 = 10 sweets, and there are 5 × 5 = 25 sweets altogether.
The rule is the same both times: divide the amount you know by the number of parts it fills. The total fills all the parts. A share fills only its own parts.
Bo's 3 parts hold 15 sweets, so every part holds 5. Amy's 2 parts are 10 sweets, and all 5 parts are 25.
The usual mistakes
Dividing by the wrong number of parts. When 2 : 3 shares 20, the 20 fills all 5 parts, so divide by 5. Dividing 20 by 2 or by 3 treats the whole amount as if it were one share.
Answering with the number of parts. 5 is how many parts there are. What one part is worth is 20 ÷ 5 = 4.
Taking one part away from the total. 20 − 4 = 16 is not what one part is worth, and it is not a share either: the shares are 8 and 12.
Worked example: A Mixture Made in Two Batch Sizes
Question An orange drink is made by mixing orange concentrate and water in the ratio 2 : 7. (a) How much water is needed to mix with 250 ml of concentrate? (b) A jug holds 3.6 liters of the drink. How much concentrate is in the jug?
1.Every batch is 2 units of concentrate and 7 units of water. With 250 ml of concentrate, 2 units = 250 ml.
The concentrate is 2 units and the water is 7 units. Here 2 units are 250 ml. 2.1 unit = 250 ÷ 2 = 125 ml.
1 unit = 250 ÷ 2 = 125 ml. 3.(a) The water is 7 units: 7 × 125 = 875 ml.
(a) The water is 7 × 125 = 875 ml. 4.In the jug the whole drink is 2 + 7 = 9 units, and 3.6 liters = 3600 ml. So 1 unit = 3600 ÷ 9 = 400 ml.
In the jug all 9 units are 3600 ml, so 1 unit = 3600 ÷ 9 = 400 ml. 5.(b) The concentrate is 2 × 400 = 800 ml. Check: 800 + 7 × 400 = 3600 ml.
(b) The concentrate is 2 × 400 = 800 ml.
Answer: (a) 875 ml; (b) 800 ml
Common mistakes
- Using 1 unit = 125 ml again in part (b). The jug is a different batch, so one unit has a new size and must be found from the 3600 ml.
- Dividing 3600 ml by 7 or by 2 in part (b). The 3600 ml is the whole drink, which is 2 + 7 = 9 units.
Three shares from two ratios
The unitary method works for any number of shares. In the next problem, two ratios are first combined into one three-part ratio, as in Combining Two Ratios. Then the total fills all the parts, and one part follows.
Worked example: Two Ratios That Share One Person
Question The ratio of Ann's savings to Ben's savings is 2 : 3. The ratio of Ben's savings to Cal's savings is 4 : 5. The three children have saved $210 altogether. (a) How much has Ben saved? (b) How much more has Cal saved than Ann?
1.Ben is 3 units in the first ratio and 4 units in the second. The lowest common multiple of 3 and 4 is 12, so make Ben 12 units in both.
Ben is 3 units in the first ratio and 4 units in the second. Make him 12 units in both. 2.Multiply the first ratio by 4: 2 : 3 = 8 : 12. Multiply the second ratio by 3: 4 : 5 = 12 : 15. So Ann : Ben : Cal = 8 : 12 : 15.
2 : 3 = 8 : 12 and 4 : 5 = 12 : 15, so Ann : Ben : Cal = 8 : 12 : 15. 3.There are 8 + 12 + 15 = 35 units altogether, so 35 units = $210 and 1 unit = 210 ÷ 35 = $6.
The 8 + 12 + 15 = 35 units are $210, so 1 unit is $6. 4.(a) Ben has saved 12 × 6 = $72.
(a) Ben has saved 12 × 6 = $72. 5.(b) Cal has 15 − 8 = 7 units more than Ann, which is 7 × 6 = $42. Check: 48 + 72 + 90 = 210.
(b) Cal has 7 units more than Ann: 7 × 6 = $42.
Answer: (a) $72; (b) $42
Common mistakes
- Joining the two ratios as 2 : 3 : 5 or 2 : 4 : 5. Ben is 3 units in one ratio and 4 units in the other, so the units are different sizes until Ben is made 12 units in both.
- Multiplying only Ben's term by 4 and leaving Ann's term as 2. Both terms of a ratio must be multiplied by the same number, or the ratio changes.