Methods and worked examples: Proportional Reasoning-1
Proportional Reasoning-122 min readLesson 2 of 5
2.1 Finding missing terms
Example 6: when Neelima was 3, her mother was 10 times her age, so the ratio was 3 : 30 = 1 : 10. Nine years later Neelima is 12 and her mother is 39, so the ratio is 12 : 39 = 4 : 13. Adding the same number to both terms changes the ratio. Ratios of ages are a common trap because age keeps changing.
Example 7: fill the missing numbers in ratios proportional to 14 : 21. For the pair with second term 42, the second term became 2 times larger, so the first term is 28: the ratio is 28 : 42. For the pair with first term 6, the scale factor is 6/14 = 3/7, and 21 × 3/7 = 9, giving 6 : 9. For the pair with first term 2, both terms were divided by 7, giving 2 : 3.
The method is always the same: find the factor that changed the known term, then apply it to the other term.
2.2 Filter coffee and the direction of the ratio
Manjunath mixes coffee decoction and milk. His regular cup uses 15 mL of decoction with 35 mL of milk, so the ratio is 15 : 35 = 3 : 7. A stronger cup needs more decoction per unit of milk, which means a larger first term for the same second term. A lighter cup needs the opposite.
The chapter's table asks whether each mixture is regular, strong or light:
| Decoction (mL) | Milk (mL) | Simplest form | Comparison |
|---|---|---|---|
| 300 | 600 | 1 : 2 | stronger than 3 : 7 |
| 150 | 500 | 3 : 10 | lighter |
| 200 | 400 | 1 : 2 | stronger |
| 24 | 56 | 3 : 7 | regular |
| 100 | 300 | 1 : 3 | lighter |
To compare ratios, put each into simplest form and compare the two fractions. The ratio 1 : 2 means half a unit of milk per unit of decoction, while 3 : 7 is about 0.43, so 1 : 2 is stronger. The ratio 1 : 3 is about 0.33, so it is lighter.
2.3 Cross multiplication and the Rule of Three
If a : b :: c : d, the two ratios change by the same factor, so ad = bc. This is called cross multiplication. It turns a proportion into one equation in which the unknown can be isolated: if three of the four quantities are known, the fourth is d = bc/a.
Ancient Indian mathematicians called such problems the Rule of Three. Āryabhaṭa named the three known quantities pramāṇa (the measure), phala (the fruit) and ichchhā (the requisition), and the unknown ichchhāphala (the yield). His rule says: multiply the fruit by the requisition and divide by the measure, that is, yield = (phala × ichchhā) / pramāṇa.
2.4 Worked examples with cross multiplication
Example 8: the cook prepares 15 kg of rice for 120 students. When a rainy day leaves just 80 students present, the quantity must fall in the same proportion. Write the relation as in the math block below. The first term changed by the factor 80/120 = 2/3, so the rice shrinks by the same factor and comes to 10 kg.
Example 9: a car covers 90 km in 150 minutes. How far does it go in 4 hours at the same speed? The units must match, so change 4 hours to 240 minutes. Then 150 : 90 :: 240 : x, and x = 240 × 90 ÷ 150 = 144 km.
Example 10: Himachal tea sells at 200 g for ₹200, so the weight to price ratio is 1 : 1 in grams and rupees. Meghalaya tea sells 1 kg for ₹800, which is 1000 : 800 = 5 : 4. The ratios are not proportional. Comparing the price of 1 kg, Himachal tea costs ₹1000 and Meghalaya tea costs ₹800, so Himachal tea is more expensive.
2.5 Sharing a quantity in a given ratio
Section 7.5 divides a quantity according to a ratio. If x is divided in the ratio m : n, there are m + n equal parts. Each part is x/(m + n), so the first share is m times the unit part and the second is n times it. Those two shares sit in the ratio m : n.
Twelve counters shared 3 : 1 give 4 parts of 3, so 9 counters and 3 counters. Forty-two counters shared 4 : 3 give 7 parts of 6, so 24 and 18.
Example 11: Prashanti invests ₹75,000 and Bhuvan invests ₹25,000, a ratio of 3 : 1. A profit of ₹4,000 is shared in the same ratio. There are 4 parts, each ₹1,000, so Prashanti gets ₹3,000 and Bhuvan gets ₹1,000.
Example 12: a 40 kg mixture has sand and cement in the ratio 3 : 1. The sand is 3/4 × 40 = 30 kg and the cement is 1/4 × 40 = 10 kg. To make the ratio 5 : 2, the sand stays at 30 kg, so cement must be 2/5 × 30 = 12 kg. There is already 10 kg of cement, so 2 kg must be added.
2.6 Unit conversions
Proportional problems often start with a unit conversion. The chapter lists the useful ones:
- Length: one metre is about 3.281 feet.
- Area: one square metre is 10.764 square feet; an acre is 43,560 square feet; a hectare is 10,000 square metres or about 2.471 acres.
- Volume: a millilitre is the same as a cubic centimetre, and a litre holds 1,000 millilitres.
- Temperature: convert with 9/5 times the Celsius value plus 32 to reach Fahrenheit, and with 5/9 times (Fahrenheit minus 32) to come back. Thus 25°C is 77°F.
Write the conversion as a proportion, cancel the units, and keep the same unit on both sides of the ratio. The next two exercise pages use these conversions in real problems.