7.8 Going Deeper: Enrichment & Exam Preparation

Where Proportional Reasoning Shows Up in Real Life

Math in the Real World

  • Scaling a recipe. A pancake batter for 8 servings uses 2 eggs and 300 mL of milk. For a party of 20, keep the ratios fixed: \frac{20}{8} = 2.5 times everything, so 2 \times 2.5 = 5 eggs and 300 \times 2.5 = 750 mL of milk. Cooking is proportional reasoning you can eat.
  • Reading a map. A map drawn to scale 1 : 50{,}000 means 1 cm on paper stands for 50{,}000 cm on the ground. A road measuring 4 cm on the map is really 4 \times 50{,}000 = 200{,}000 cm = 2 km long. Every map, floor plan and model car runs on a ratio.
  • Changing money. If 1 US dollar trades for about ₹83, then a \$250 gift converts to 250 \times 83 = ₹20{,}750. Exchange rates are just proportions between two currencies.
  • Speed as a rate. A train covering 540 km in 9 hours travels at 540 \div 9 = 60 km/h, a “per-hour” unit rate. At that steady speed it would cover 60 \times 5 = 300 km in 5 hours. Speed, typing speed, heart rate and price-per-kg are all unit rates.
  • Mixing and diluting. A nimbu-paani recipe of 1 part syrup to 4 parts water keeps the same taste whether you make a glass or a bucket, provided the ratio stays 1 : 4. Cement-sand-aggregate concrete (1 : 2 : 4) and paint shades work the very same way.

Quick Reference: Ratio, Proportion & Conversions

Knowing these facts and steps by heart turns slow word problems into quick, confident work, a real advantage under exam time pressure.

Idea What it means Example
Ratio a : b for every a of one, b of the other 80 : 60 (width : height)
Simplest form divide both terms by their HCF 80 : 60 = 4 : 3
Proportion a:b::c:d the two ratios are equal 5 : 6 :: 60 : 72
Cross multiplication a:b::c:d \iff ad = bc 5\times72 = 6\times60 = 360
Missing term d = \dfrac{bc}{a} (Rule of Three) 7:12::21:x \Rightarrow x = 36
Share x in m:n each part = \dfrac{x}{m+n} 4500 in 4:5 \Rightarrow 2000, 2500
Unitary-method step What you do
1. Find the rate for one divide to get the value of a single unit
2. Scale up to many multiply that rate by the new quantity
Example 9 books cost ₹270 \Rightarrow one book ₹30 \Rightarrow 13 books ₹390
Quantity Conversion
Length 1 m = 100 cm; \;1 km = 1000 m
Mass 1 kg = 1000 g; \;1 tonne = 1000 kg
Volume 1 L = 1000 mL; \;1 mL = 1 cc
Time 1 h = 60 min; \;1 day = 24 h
Temperature F = \tfrac{9}{5}C + 32; \;C = \tfrac{5}{9}(F-32)

Use the best-buy comparator to turn two “price for a quantity” deals into a common unit rate and see which is cheaper.

Best-buy / unit-rate comparator
Deal A: price for units
Deal B: price for units

Memory Tricks & One-Page Revision

Quick Revision Card

  • Ratio means division, not subtraction. Compare by how many times (\div), never by how much more (-).
  • Simplest form: divide both terms by the HCF. 84 : 144 = 7 : 12.
  • Proportion test: a:b::c:d exactly when ad = bc (cross products equal).
  • Find the missing term: d = \dfrac{bc}{a}, “phala \times icchā \div pramāṇa.”
  • Unitary method: first find the value of one, then multiply for many.
  • Sharing in m:n: total parts = m+n; one part = \dfrac{x}{m+n}; shares are m and n parts.
  • Same units first! Convert minutes↔︎hours, g↔︎kg, cm↔︎m before you set up the proportion.
  • Watch for inverse cases: faster speed → less time. That is not a direct proportion.

Spot the Mistake

Common Exam Mistakes

  • Comparing by difference. Saying 80:60 and 60:40 are “the same shape” because both drop by 20, wrong; the factors (\tfrac34 vs \tfrac23) differ.
  • Forgetting to convert units. Writing 180 : 120 :: 5 : x with 180 in minutes but 5 in hours. Convert 5 h = 300 min first.
  • Flipping the ratio. In a:b::c:x, the matching quantity must sit in the same position; mixing km : min with min : km gives nonsense.
  • Sharing wrongly. For \$4500 in 4:5, dividing by 2 instead of by 4+5 = 9 parts. Each part is 4500 \div 9 = 500.
  • Treating speed–time as direct. Doubling the speed does not double the time, it halves it. That is inverse proportion.

Test Your Reflexes

A fast, friendly drill: a small proportion appears with one term missing, and you type the missing value. Build a streak!

Proportion reflex
Find x so the proportion holds, then press Check.
3 : 4 :: 9 : x
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. The ratio 84 : 144 in its simplest form is ______.
  2. If 7 : 12 :: 21 : x, then x = ______.
  3. When ₹4500 is shared in the ratio 4 : 5, the smaller share is ______.

Short answer (2 marks each)

  1. Find the missing term: 5 : 8 :: \underline{\ \ } : 56.
  2. If 9 identical books cost ₹270, find the cost of 13 such books using the unitary method.

Long answer (3 marks each)

  1. Divide ₹7200 among three children in the ratio 3 : 4 : 5. Verify that the shares add up to ₹7200.
  2. A tank holds 90 L of a milk–water mixture in the ratio 7 : 2. How much water must be added to change the ratio to 7 : 3?

HOTS (Higher Order Thinking)

  1. Two numbers are in the ratio 5 : 7 and their sum is 96. Find the two numbers.
  2. A car covers 60 km in 90 minutes; a scooter covers the same 60 km in 120 minutes. Find the ratio of their speeds, and explain why the faster vehicle has the larger ratio term.

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): 0.4 : 0.5 in simplest form is 4 : 5.   Reason (R): Multiplying both terms of a ratio by the same non-zero number leaves the ratio unchanged.
  1. HCF of 84 and 144 is 12: 84 : 144 = 7 : 12.
  2. Cross-multiply: 7x = 12 \times 21 = 252, so x = \mathbf{36}.
  3. Parts = 4 + 5 = 9; each part = 4500 \div 9 = 500. The smaller share is 4 \times 500 = \mathbf{₹2000}.
  4. Cross-multiply: 8 \times \underline{\ \ } = 5 \times 56 = 280, so the term is 280 \div 8 = \mathbf{35}.
  5. One book costs 270 \div 9 = ₹30; so 13 books cost 13 \times 30 = \mathbf{₹390}.
  6. Parts = 3 + 4 + 5 = 12; each part = 7200 \div 12 = 600. Shares: 3 \times 600 = ₹1800, 4 \times 600 = ₹2400, 5 \times 600 = ₹3000. Check: 1800 + 2400 + 3000 = \mathbf{₹7200}. ✓
  7. With 7 + 2 = 9 parts in 90 L, each part = 10 L, so milk = 70 L and water = 20 L. The milk stays 70 L. For the new ratio 7 : 3, we need water = \dfrac{70}{7} \times 3 = 30 L. So add 30 - 20 = \mathbf{10\text{ L}} of water.
  8. Parts = 5 + 7 = 12; each part = 96 \div 12 = 8. The numbers are 5 \times 8 = \mathbf{40} and 7 \times 8 = \mathbf{56} (and 40 + 56 = 96). ✓
  9. Car speed = 60 \div \tfrac{90}{60} = 60 \div 1.5 = 40 km/h; scooter speed = 60 \div \tfrac{120}{60} = 60 \div 2 = 30 km/h. Ratio = 40 : 30 = \mathbf{4 : 3}. The faster vehicle covers more distance per hour, so its speed value, and hence its ratio term, is larger.
  10. (a), Both true and R explains A: multiplying 0.4 : 0.5 by 10 gives 4 : 5, which is exactly the rule that scaling both terms by the same number preserves the ratio.

Connections to Other Chapters

How This Chapter Links Forward

  • Chapter 8 (Fractions in Disguise): a ratio a : b is really the fraction \tfrac{a}{b} in disguise, so reducing ratios and reducing fractions are the same skill.
  • Chapter 10 (Proportional Reasoning-2): the inverse proportions hinted at here (speed and time, workers and days) are studied in full, including percentages and direct-vs-inverse tests.
  • Chapter 14 (Area): map scales and the unit conversions of length and area used here reappear when measuring real land and figures.

Glossary

Key Terms

  • Ratio (a : b): a comparison of two quantities by division; “for every a of one there are b of the other.”
  • Terms: the two numbers a and b that make up a ratio.
  • Simplest form: a ratio with its terms divided by their HCF, so they share no common factor.
  • Proportion (a : b :: c : d): a statement that two ratios are equal.
  • Cross multiplication: the test ad = bc that decides whether a:b::c:d.
  • Rule of Three (Trairāśika): finding a missing fourth term by d = \tfrac{bc}{a}.
  • Unitary method: find the value of one unit first, then scale to the required number.
  • Unit rate: a quantity expressed “per one” (per kg, per hour), used to compare deals or speeds.