Sec 1 Ratio, Rate and Proportion: Methods & Examples

Secondary 1 Mathematics · G3 focus · Study guide

Compare quantities in the same units, share totals, join ratios and track changes using clear bar models.

Concept → worked examples → practice18 min read · practice linked below

A ratio compares equal-sized parts

If red beads and blue beads are in the ratio $2:3$ , every group of 2 red beads matches a group of 3 blue beads. There are $2+3=5$ parts altogether, so red beads make up $\frac25$ of all the beads. The ratio red to blue is $\frac23$ ; the fraction of the total that is red is $\frac25$ . Read the order carefully.

Equivalent ratios multiply or divide every term by the same positive number: $2:3=4:6=10:15$ . To simplify, first use the same units, clear any fractions or decimals, then divide by the HCF. A ratio comparing like quantities has no unit once those units cancel.

Use the sum of the ratio terms when the given quantity is a total.

Worked example 1: units and fractional ratios

Simplify $0.75\text{ kg}:300\text{ g}$ , and then simplify $\frac23:\frac56$ .

Convert kilograms to grams: $0.75\text{ kg}=750\text{ g}$ . The first ratio becomes $750:300$ .
The HCF is 150, so divide both terms by 150: $750:300=\boxed{5:2}.$
For the second ratio, multiply both fractions by 6, the LCM of the denominators: $\left(6\times\frac23\right):\left(6\times\frac56\right)=\boxed{4:5}.$

Check: $750/300=5/2$ , and $(2/3)\div(5/6)=4/5$ . A ratio is preserved by common multiplication, not by adding the same number to each term.

Introduce one part $k$ . If $A:B:C=2:3:4$ , then $A=2k$ , $B=3k$ , $C=4k$ . A total corresponds to $9k$ , while the difference between $C$ and $A$ is $2k$ . Label what the given quantity represents before dividing.

Worked example 2: divide a total in three parts

Share S$180 among Asha, Ben and Chen in the ratio $2:3:4$ .

The total represents $2+3+4=9$ parts.
One part is $180\div9=20$ dollars.
Asha receives $2(20)=\boxed{\text{S\$40}}$ , Ben receives $3(20)=\boxed{\text{S\$60}}$ , and Chen receives $4(20)=\boxed{\text{S\$80}}$ .

Check: $40+60+80=180$ , and dividing each share by 20 recovers $2:3:4$ . Asha's fraction of the total is $2/9$ , not $2/3$ .

Worked example 3: join two ratios using the shared quantity

Given $A:B=2:3$ , $B:C=4:5$ , and C has 21 more counters than A, find all three quantities.

The B terms must match. Their LCM is 12: $A:B=8:12,\qquad B:C=12:15.$
Therefore $A:B:C=8:12:15$ . The difference $C-A$ is $15-8=7$ parts.
One part is $21\div7=3$ counters. Thus $\boxed{A=24,\quad B=36,\quad C=45}.$

Check: $24:36=2:3$ , $36:45=4:5$ , and $45-24=21$ . Joining the original ratios as $2:3:5$ would assign two different sizes to B.

Explore a fixed ratio

Start with the ratio $A:B=2:3$ . Change the total to see why each share changes but its fraction of the total stays constant. This model shares a continuous quantity such as a length of ribbon.

Total 50 m; ratio 2 : 3. One part = 10 m. A = 20 m and B = 30 m.

The bar shows the shares of a positive total. Both shares scale together.

For indivisible items, a result such as 2.5 counters signals that the chosen total cannot be shared in that exact ratio.

When a ratio changes, track what stays the same

Transferring an amount from one person to another preserves the total. Adding an amount from outside changes the total. The “one part” in the new ratio need not equal one part in the old ratio.

The decrease and increase cancel:

-4+4=0

Worked example 4: transfer between two containers

Water in containers A and B is initially in the ratio $3:2$ . After 4 litres is poured from A into B, the ratio is $1:1$ . Find the original amounts.

Let the original amounts be $3k$ and $2k$ litres. After the transfer they are $3k-4$ and $2k+4$ .
The new amounts are equal, so $3k-4=2k+4\quad\Rightarrow\quad k=8.$
Originally, A contained $3(8)=\boxed{24\text{ litres}}$ and B contained $2(8)=\boxed{16\text{ litres}}$ .

Check: after transferring 4 litres, both contain 20 litres. The total remains 40 litres. The original difference was 8 litres because the transfer changes both amounts.

A unit rate compares different units

A ratio such as $2:3$ compares quantities in a fixed order. A rate tells how much of one quantity corresponds to one unit of another: S$ per kilogram, litres per minute, or kilometres per hour. Keep units attached to a rate. With a constant unit rate, multiply by the number of units to find the total.

Worked example 5: choose the lower unit price

Rice costs S$6.30 for a 1.5 kg bag or S$4.60 for a 1 kg bag. Which is cheaper per kilogram? What would 3 kg cost using each bag size?

For the 1.5 kg bag, $\text{price per kg}=6.30\div1.5=\text{S\$4.20/kg}.$
The 1 kg bag costs S$4.60/kg. Thus the 1.5 kg bag is cheaper per kilogram by $4.60-4.20=\text{S\$0.40/kg}$ .
For 3 kg, two large bags cost $2(6.30)=\text{S\$12.60}$ ; three small bags cost $3(4.60)=\text{S\$13.80}$ . The saving is $\boxed{\text{S\$1.20}}$ .

Check: $3\times0.40=1.20$ . Compare the same quantity and take account of whole packs when deciding what to buy.

For currency exchange, the quoted rate also tells the direction. At a hypothetical rate of S$1 = 0.80 foreign dollars, S$50 converts to $50(0.80)=40$ foreign dollars. To convert 40 foreign dollars back, divide by 0.80. Actual fees and quoted rates would need to be considered in a real purchase.

Common mistakes

Different units: 1 m : 50 cm is $100:50=2:1$ , not $1:50$ .
Confusing part-to-part and part-to-whole: in $2:3$ , the first part is $2/5$ of the total.
Using a total when given a difference: subtract ratio terms for a difference.
Matching the wrong middle term: scale both ratios so that the shared quantity agrees.
Keeping the old part size after a transfer: define actual amounts and form the new relationship.

Continue with the question bank

Try ratio, rate and proportion questions with worked solutions. For rates involving journeys and time, continue to Rate, Speed and Time.