Chapter focus
Ratios compare quantities multiplicatively. Equivalent ratios, unit rates and proportional relationships help solve sharing, scaling and price problems.
These original notes support revision. Refer to the full activities and exercise sequence in your assigned Ganita Prakash edition.
1. Ratios compare quantities
A ratio compares two quantities in a stated order. If there are 12 red counters and 8 blue counters, the ratio of red to blue is 12:8 = 3:2 after dividing both terms by 4. Reversing the order gives blue to red = 2:3.
Before comparing measures, use compatible units. A ratio of 2 metres to 50 centimetres is not 2:50 in simplest comparable form: convert 2 m to 200 cm, giving 200:50 = 4:1.
Ratios can be written as a:b or as a fraction a/b (when b ≠ 0). The ratio 3:5 describes 3 parts to 5 parts; it does not mean 3 out of 5 unless the quantities are parts of a total of 8 equal parts.
2. Equivalent ratios and scaling
Multiply or divide both terms of a ratio by the same non-zero number to make an equivalent ratio: 3:5 = 6:10 = 9:15. The relationship between the compared quantities stays the same.
Worked example: share ₹560 in the ratio 3:5
- Total parts = 3 + 5 = 8.
- Value of one part = ₹560 ÷ 8 = ₹70.
- First share = 3 × ₹70 = ₹210; second share = 5 × ₹70 = ₹350.
- Check: ₹210 + ₹350 = ₹560 and 210:350 simplifies to 3:5.
3. Unit rates make comparisons fair
A unit rate gives the amount for one unit. If 4 notebooks cost ₹120, the price per notebook is ₹120 ÷ 4 = ₹30. Unit rates let us compare offers of different sizes.
For example, 6 pens for ₹90 cost ₹15 per pen. An offer of 8 pens for ₹136 costs ₹17 per pen. If the quality and conditions are the same, the first offer has the lower unit price.
4. Direct proportion
Two quantities are directly proportional when one is a fixed multiple of the other. If each notebook costs ₹30, then cost = 30 × number of notebooks. The ratio of cost to notebooks is constant, and doubling the number doubles the cost.
Worked example: scale a recipe
A recipe uses 2 cups of rice for 5 people. For 15 people, the scale factor is 15 ÷ 5 = 3, so rice needed = 2 × 3 = 6 cups. This assumes the same serving size and no change to the recipe.
Not every real situation is directly proportional. A taxi fare with a fixed starting charge does not double when the distance doubles, because part of the cost is fixed.
5. Solve proportions carefully
If a/b = c/d with b and d non-zero, then ad = bc. This cross-product check can solve for an unknown, but first confirm that the quantities really have a proportional relationship and their units correspond.
Worked example
At a steady rate, 3 kg of apples cost ₹240. Let the cost of 5 kg be ₹x. Then x/240 = 5/3, so 3x = 1,200 and x = ₹400. The unit price check gives ₹80 per kg in both cases.
Common errors to avoid
- Keep the order of a ratio consistent: girls:boys is not boys:girls.
- Convert to the same units before simplifying a ratio.
- In a part-to-part ratio 3:5, the total is 8 parts, not 5.
- Check whether a fixed rate assumption is reasonable before scaling.
Practice: show the relationship
1. Simplify 18:24.
Answer: Divide both terms by 6: 3:4.
2. A recipe uses 2 cups of rice for 5 people. How much is needed for 15 people?
Answer: The scale factor is 15/5 = 3; 2 × 3 = 6 cups.
3. Which is cheaper per litre: 2 L for ₹110 or 5 L for ₹260?
Answer: ₹55/L versus ₹52/L. The 5 L offer is cheaper per litre.
4. A cyclist travels 36 km in 2 hours at a steady pace. How far in 5 hours?
Answer: Rate = 36 ÷ 2 = 18 km/h. Distance = 18 × 5 = 90 km, assuming the pace stays steady.
5. Divide 720 marbles between two teams in the ratio 5:7.
Answer: There are 12 parts, each worth 60. The shares are 300 and 420 marbles.
Chapter summary
Ratios compare quantities in order; equivalent ratios preserve the relationship. Unit rates make comparisons easier, while direct proportion models situations with a constant multiplicative rate. Check units, assumptions and totals in every solution.