IP Maths Notes (Lower Sec, Year 1-2): 03) Ratio, Rate & Percentage Reasoning

Mastery of ratio and percentage reasoning saves marks across chemistry mixtures, physics rates, and finance contexts. This post trains systematic setups and calculator-free checks.

Learning targets

• Simplify ratios and express them in different forms (part:part, part:whole).
• Convert between rates (speed, density, unit pricing) with clear units.
• Chain multiple percentage changes and interpret reverse percentages.
• Model mixtures and part-whole distributions using algebraic reasoning.

1. Ratio fundamentals

1.1 Simplification & scaling

• Divide all parts by the highest common factor to simplify.
• To increase a ratio, multiply each part by the same scale factor.

Example: The ratio of red:blue marbles is 18:24. Simplify: divide by 6 → 3:4.

1.2 Part-whole conversion

If the ratio of boys to girls is 5:4, the total number of parts is 9. Boys make \( \frac{5}{9} \) of the class.

2. Rate conversions

Common rate templates:

• Speed: \( \text{speed} = \frac{\text{distance}}{\text{time}} \).
• Density: \( \text{density} = \frac{\text{mass}}{\text{volume}} \).
• Unit cost: \( \text{cost per unit} = \frac{\text{total cost}}{\text{quantity}} \).

Worked example — Multi-rate problem

A cyclist travels 18 km in 40 minutes on level ground and 12 km in 35 minutes uphill. What is the average speed for the combined journey?

Total distance: \( 18 + 12 = 30 \text{ km} \).

Total time: \( 40 + 35 = 75 \text{ minutes} = 1.25 \text{ h} \).

Average speed: \( \frac{30}{1.25} = 24 \text{ km h}^{-1} \).

3. Percentage techniques

3.1 Successive percentage changes

To apply a 20% increase then a 15% decrease to an amount \( A \):

\[ A \times 1.20 \times 0.85 = 1.02A. \]

Net effect is a 2% increase.

3.2 Reverse percentages

If an item costs \(\$204 \) after a 15% discount, original price \( P \) satisfies \( 0.85P = 204 \).

\[ P = \frac{204}{0.85} = 240. \]

3.3 Percentage point vs percentage change

• Percentage points describe absolute difference (e.g., 65% to 70% is +5 percentage points).
• Percentage change compares relative growth: \( \frac{70 - 65}{65} \times 100\% \approx 7.69\% \).

4. Mixture models

4.1 Part-part mixture

To combine two solutions of different concentrations:

• Let \( m_1, m_2 \) be masses (or volumes).
• Let \( c_1, c_2 \) be concentrations.
• Resulting concentration: \( \frac{m_1 c_1 + m_2 c_2}{m_1 + m_2} \).

Worked example — Alloy blend

A jeweller mixes 120 g of 18-karat gold (75% pure) with 80 g of 14-karat gold (58.3% pure). What is the purity of the alloy?

Mass of pure gold: \( 120 \times 0.75 + 80 \times 0.583 = 90 + 46.64 = 136.64 \text{ g} \).

Total mass: \( 200 \text{ g} \).

Purity: \( \frac{136.64}{200} = 0.6832 = 68.32\% \).

Try it yourself

1. Siobhan and Amir share money in the ratio 7:9. If Amir has \(\$63 \), how much does Siobhan have?
2. A car travels at 72 km h\( ^{-1} \) for 35 minutes then at 54 km h\( ^{-1} \) for 50 minutes. Find the average speed for the journey.
3. The population of a town grows by 12% one year and shrinks by 5% the next. What is the net percentage change?
4. A 250 ml drink is 8% sugar. How much 15% sugar syrup must be added to reach a 10% mixture?

Continue with graph work at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-04-Coordinate-Geometry-and-Graphs.