Dividing Quantity into Given Ratio
Dividing a quantity in a given ratio means sharing it so that each share matches its number of parts. To share \(\$72\) in the ratio \(4:5\), there are \(9\) parts, so one part is \(\$8\) and the shares are \(\$32\) and \(\$40\). In NSW Year 8 maths this unitary method handles every case (NESA outcome MA4‑RAT‑C‑01).
You will share amounts in ratios with two, three or four terms, including decimal and fraction ratios, mixed units and money to the cent. You will work backwards from one share, or from the difference between two shares, to find the total. Harder problems hand out a fixed amount first, move marbles from one person to another, or split the angles of a quadrilateral. Garage-sale earnings and concrete mixes are typical. It is part of Year 8 Rates and Ratios.
Theory
To divide a quantity in a ratio, add the terms to get the total number of parts, find the value of one part, then multiply by each term. \(1200\) cm in the ratio \(3:4:5\) has \(12\) parts of \(100\) cm, so the pieces are \(300\), \(400\) and \(500\) cm. This Year 8 maths page shows how.
The three steps (the unitary method): (1) total number of parts \(=\) add the terms; (2) one part \(=\) quantity \(\div\) total parts; (3) each share \(=\) its number of parts \(\times\) one part. Check that the shares add up to the total.
Before you start, change every amount to the same unit and make the ratio whole numbers: multiply by \(10\) for decimals, or by the lowest common denominator for fractions.
Money. One part need not be a whole number of dollars. Keep its exact value and write each share to \(2\) decimal places.
One share known: one part \(=\) that share \(\div\) its number of parts. Then total \(=\) total parts \(\times\) one part.
A difference known: one part \(=\) the difference \(\div\) the difference of the two terms.
The shares known: write them as a ratio and simplify. An equal share is half of the total number of parts.
Extra steps. If a fixed amount is handed out first, take it off the total and share what is left. If one person passes part of their share to another, the total does not change, so divide the same total by the new number of parts.
Which number do you divide, and by what?
| You are given | One part = |
|---|---|
| the total | total ÷ (sum of the terms) |
| one share | that share ÷ its term |
| the difference of two shares | difference ÷ (difference of their terms) |
Dividing a quantity in a ratio
- Tidy up: same units, whole-number ratio.
- Add the terms for the total number of parts.
- Divide the quantity by the total number of parts to find one part.
- Multiply one part by each term.
- Check that the shares add up to the quantity.
| \(\text{Parts}\) | \(=\) | \(3 + 4 + 5 = 12\) |
| \(\text{One part}\) | \(=\) | \(1200 \div 12 = 100\) |
| \(3 \times 100\) | \(=\) | \(300\) |
| \(4 \times 100\) | \(=\) | \(400\) |
| \(5 \times 100\) | \(=\) | \(500\) |
\(300\) cm, \(400\) cm and \(500\) cm. Check: \(300 + 400 + 500 = 1200\).
| \(\text{Difference in parts}\) | \(=\) | \(9 - 5 = 4\) |
| \(\text{One part}\) | \(=\) | \(20 \div 4 = 5\) |
| \(\text{Smaller}\) | \(=\) | \(5 \times 5 = 25\) |
The smaller number is \(25\) (and the larger is \(45\): \(45 - 25 = 20\)).
| \(\text{One part}\) | \(=\) | \(58.80 \div 8 = 7.35\) |
| \(3 \times 7.35\) | \(=\) | \(22.05\) |
| \(4 \times 7.35\) | \(=\) | \(29.40\) |
\(\$7.35\), \(\$22.05\) and \(\$29.40\). Check: they add to \(\$58.80\).
\(1.5:2:2.5 = 3:4:5\) \((\times 2)\), and \(2.4\) kg \(= 2400\) g.
| \(\text{One part}\) | \(=\) | \(2400 \div 12 = 200\) |
| \(3 \times 200\) | \(=\) | \(600\) |
| \(4 \times 200\) | \(=\) | \(800\) |
| \(5 \times 200\) | \(=\) | \(1000\) |
\(600\) g, \(800\) g and \(1000\) g.
(i) What were the takings?
(ii) How much goes to equipment?
| \(\text{One part}\) | \(=\) | \(186 \div 3 = 62\) |
| \(12 \times 62\) | \(=\) | \(744\) |
| \(4 \times 62\) | \(=\) | \(248\) |
(i) \(\$744\)
(ii) \(\$248\). Check: juniors \(5 \times 62 = 310\), and \(310 + 186 + 248 = 744\).
(ii) Two sisters get \(\$15\) each from \(\$150\), and the rest is shared in the ratio \(2:3\). How much does each get altogether?
(i) Multiply by \(6\): the ratio is \(2:1:3\), \(6\) parts.
(ii) Take off the fixed amounts first: \(150 - 30 = 120\) is shared in \(5\) parts.
| \(90 \div 6\) | \(=\) | \(15\) |
| \(120 \div 5\) | \(=\) | \(24\) |
| \(2 \times 24 + 15\) | \(=\) | \(63\) |
| \(3 \times 24 + 15\) | \(=\) | \(87\) |
(i) \(\$30\), \(\$15\) and \(\$45\)
(ii) \(\$63\) and \(\$87\)
Cards only move, so the total stays \(120\).
| \(\text{Ravi before}\) | \(=\) | \(5 \times 120 \div 8 = 75\) |
| \(\text{Ravi after}\) | \(=\) | \(1 \times 120 \div 3 = 40\) |
| \(75 - 40\) | \(=\) | \(35\) |
Ravi gave Dan \(35\) cards. Check: Dan \(45 + 35 = 80\), and \(40:80 = 1:2\).
There are \(5\) parts, so an equal share is \(2.5\) parts: \(0.5\) of a part more than \(2\).
| \(\text{One part}\) | \(=\) | \(10 \div 0.5 = 20\) |
| \(\text{Total}\) | \(=\) | \(5 \times 20 = 100\) |
\(\$100\). Check: shares \(\$40\) and \(\$60\); half is \(\$50\), which is \(\$10\) more than \(\$40\).
Common pitfalls
Frequently asked questions
How do I divide a quantity in a ratio?
Add the terms, divide the quantity by that total to get one part, then multiply one part by each term.
What if I know one share but not the total?
Divide that share by its term to get one part, then multiply by the total number of parts.
What if I know the difference between two shares?
Divide the difference by the difference of the two terms to get one part.
How do I divide in a ratio of fractions?
Multiply every term by the lowest common denominator to make whole numbers first.
How do I check my answer?
Add the shares; they must make the original quantity.