Coaching Centre
Ratios and dividing in a ratio
Simplifying ratios with mixed units, two- and three-part ratios, dividing a quantity in a given ratio, and the unitary method when one share or a difference is known.
Builders mix concrete "one, two, three", families split a bill "in the ratio of what we earn", and a recipe for $4$ has to feed $10$. Every ratio problem has the same engine underneath: find the size of one part, then multiply.
Revise: simple ratios and map scales are in Year 11 Units, rates and conversions.
Simplifying ratios
A ratio compares quantities of the same kind, so both parts must be in the same units before you simplify. Then divide every part by the highest common factor.
Simplify $1.5$ m $: 75$ cm.
Dividing in a ratio
1. Total parts $= a + b + c$
2. One part $= \text{total} \div \text{total parts}$
3. Each share $=$ its number of parts $\times$ one part
Why does this work? A ratio of $1 : 2 : 3$ means the whole is cut into $6$ equal parts, with $1$ for the first, $2$ for the second and $3$ for the third. Once you know how big one part is, everything else follows.
Concrete is mixed from cement, sand and gravel in the ratio $1 : 2 : 3$. How much of each is in $360$ kg of dry mix?
Check Example 2 on Concrete. Then change the mix to $1 : 3 : 3$ and predict the new amount of gravel before you look. On Prize money, find the split of $\$1200$ in the ratio $5 : 3 : 2$. Finally, on Mortar, find the total mix that would use exactly $40$ kg of cement.
When you don't know the total
Sometimes you know one share, or the difference between two shares, instead of the total. Work out what that amount is in parts, divide to find one part, then carry on as before. This is the unitary method.
Profit is shared between Sam and Priya in the ratio $2 : 3$. Sam receives $\$480$. How much does Priya receive, and what was the total profit?
The ages of two cousins are in the ratio $3 : 5$, and one is $12$ years older than the other. How old are they?
Your turn
Mortar is mixed from cement and sand in the ratio $1 : 4$. A builder has three $20$ kg bags of cement. How much sand is needed, and how much mortar mix will that make?
Show the working
Cement is $1$ part $= 3 \times 20 = 60$ kg. Sand is $4$ parts $= 240$ kg. Total mix $= 5$ parts $= 300$ kg.
Simplify $1.5$ m $: 75$ cm.
$150 \text{ cm} : 75 \text{ cm}$, and dividing by $75$ gives $2 : 1$.
How many parts are there in the ratio $1 : 2 : 3$?
$1 + 2 + 3 = 6$.
$360$ kg of concrete mix is in the ratio cement : sand : gravel $= 1 : 2 : 3$. How many kg of gravel?
One part $= 360 \div 6 = 60$ kg. Gravel $= 3 \times 60 = 180$ kg.
$\$1200$ is shared in the ratio $5 : 3 : 2$. What is the middle share, in dollars?
$5 + 3 + 2 = 10$ parts, so one part $= \$120$. The middle share $= 3 \times 120 = 360$.
Profit is shared in the ratio $2 : 3$. The smaller share is $\$480$. What is the larger share, in dollars?
$2$ parts $= 480$, so $1$ part $= 240$. Larger $= 3 \times 240 = 720$.
Two ages are in the ratio $3 : 5$ and differ by $12$ years. What is the older age?
Difference $= 2$ parts $= 12$, so $1$ part $= 6$. Older $= 5 \times 6 = 30$.
Cordial is mixed $1 : 4$ with water. How much water goes with $300$ mL of cordial?
Cordial is $1$ part $= 300$ mL, so water is $4 \times 300 = 1200$ mL. ($1500$ mL is the total drink.)
A ratio $4 : 6 : 10$ in simplest form is:
Divide every part by the HCF, $2$: $2 : 3 : 5$. The decimal version is equivalent but not in simplest whole-number form.
Exam style: Mortar is cement : sand $= 1 : 4$. How many kg of sand go with three $20$ kg bags of cement?
Cement $= 60$ kg $= 1$ part. Sand $= 4 \times 60 = 240$ kg.
Exam style: How many kg of mortar mix does that make in total?
Total $= 5$ parts $= 5 \times 60 = 300$ kg. Check: $60 + 240 = 300$ ✓