Ratios & Proportions • Topic 2 of 4

Dividing in a Ratio

To split a total in a given ratio, add the ratio parts to get the number of equal shares, then hand out that many. If $8,000 is divided in the ratio 3 : 5, there are 3 + 5 = 8 shares, each worth 8000/8 = $1,000, so the two amounts are $3,000 and $5,000. The single most useful number here is the "value of one part": once you know it, every share is a quick multiply. The GRE often gives you a difference instead of a total — "the shares differ by $2,000" — and the same idea handles it: the difference is 5 − 3 = 2 parts, so one part is $1,000. Watch the order of the answer choices and answer exactly what is asked (the larger share, the difference, or the total). For three-way splits, the method is identical: divide the total by the sum of all parts.

✅ Solved examples

1. Divide $720 between two people in the ratio 4 : 5.
Shares = 4 + 5 = 9 parts. One part = 720/9 = $80. Amounts: 4×80 = $320 and 5×80 = $400.
2. A 60-litre mix is water and syrup in the ratio 7 : 3. How much syrup is in it?
Parts = 10, one part = 60/10 = 6 L. Syrup = 3 × 6 = 18 L.
3. Three friends split a profit in the ratio 2 : 3 : 5. The largest share is $1,500. Find the total.
Largest = 5 parts = 1500, so one part = $300. Total = (2+3+5) × 300 = 10 × 300 = $3,000.
4. A sum is divided in the ratio 5 : 8, and the two shares differ by $1,200. Find the total sum.
Difference = 8 − 5 = 3 parts = 1200, so one part = $400. Total = 13 × 400 = $5,200.

✏️ Practice — try these, take hints as needed

1. Divide 96 sweets between two children in the ratio 5 : 7. How many does each get?
Add the parts.
5 + 7 = 12 shares.
One part = 96/12 = 8.
40 and 56
2. $1,050 is shared in the ratio 2 : 3 : 5. Find the smallest share.
Total parts = 10.
One part = 1050/10.
Smallest = 2 parts.
$210
3. A 45 kg alloy is copper and zinc in the ratio 4 : 5. Find the mass of copper.
Parts = 9.
One part = 45/9 = 5 kg.
Copper = 4 parts.
20 kg
4. Two shares are in the ratio 3 : 7 and differ by $2,400. Find the larger share.
Difference = 7 − 3 = 4 parts.
One part = 2400/4 = 600.
Larger = 7 parts.
$4,200
5. A prize is split among three winners as 6 : 4 : 5. If the total is $4,500, what does the middle winner receive?
Total parts = 15.
One part = 4500/15 = 300.
Middle share = 4 parts.
$1,200

📝 Topic test — 8 questions

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