10.7 Going Deeper: Enrichment & Exam Preparation

Where Proportional Reasoning Shows Up in Real Life

Math in the Real World

  • Reading a map. Every atlas, road sign and trekking map carries a scale. With RF 1 : 50{,}00{,}000, a 7 cm gap on paper means 7 \times 50 = 350 km on the ground, proportion turns a ruler into a distance-measuring tool.
  • Scaling a recipe. A dosa batter at rice : urad = 3 : 1 keeps its taste whether you cook for two people or two hundred, as long as every ingredient changes by the same factor. Cooks, bakers and chemists all live by this rule.
  • Gears, speed and time. On a fixed route, speed and time are in inverse proportion: double the speed and the journey halves, because speed \times time = distance stays constant. The same logic links the teeth on two meshing gears to how fast each one spins.
  • Work rate. Two cooks chopping onions, three pumps filling a tank, or extra workers on a wall, more hands means less time, with the product (workers \times days) holding steady at the size of the job.
  • Pie charts everywhere. Election results, household budgets, battery usage and survey data are all shown as slices of 360°, each angle proportional to its share of the whole.

Exam Tip Before turning data into a pie chart, simplify the ratio by dividing every term by their HCF. The data 14 : 10 : 8 : 6 : 2 becomes 7 : 5 : 4 : 3 : 1 (dividing by 2), so each “part” is worth 360° \div 20 = 18° instead of fiddling with twenties. Smaller numbers mean fewer slips, and always finish by checking your slice angles add to exactly 360°.

Quick Reference: Scale, Proportion & Pie Charts

Keeping these facts at your fingertips turns long word problems into quick, confident steps.

Idea Rule / Formula Quick example
Representative Fraction (RF) \text{RF} = \dfrac{\text{map distance}}{\text{ground distance}} 1 : 50{,}00{,}000
RF to km 1 km = 1{,}00{,}000 cm, so divide the second term by 1{,}00{,}000 50{,}00{,}000 cm = 50 km
Map \to ground ground = map distance \times scale 7 cm \times 50 = 350 km
Larger second number in RF smaller scale, bigger area, less detail 1 : 1{,}00{,}00{,}000 (a country)
Sharing in a ratio each part = x \times \dfrac{\text{term}}{\text{sum of terms}} 500 in 5:3:2 \to 250, 150, 100
Pie-chart angle angle = \dfrac{\text{part}}{\text{total}} \times 360° \dfrac{14}{40}\times 360° = 126°
Feature Direct proportion Inverse proportion
As x goes up, y … goes up goes down
What stays constant the ratio \dfrac{x}{y} = k the product xy = k
Key equation \dfrac{x_1}{x_2} = \dfrac{y_1}{y_2} x_1 y_1 = x_2 y_2
Multiply x by n y is multiplied by n y is multiplied by \dfrac{1}{n}
Typical examples cloth \& price, parts \& paint speed \& time, workers \& days

Use the converter below to turn any map distance into a real ground distance using the scale’s second number.

Map-scale distance converter

Memory Tricks & One-Page Revision

Quick Revision Card

  • Proportional ratios: cross-multiply, a:b :: c:d means a\times d = b\times c.
  • RF on a map: 1 cm on the map = (second term) cm on the ground. Divide that second term by 1{,}00{,}000 to read it in km.
  • Bigger RF number \Rightarrow smaller scale (wider area, less detail).
  • Share x in a:b:c: add the terms, find one part = \dfrac{x}{a+b+c}, then multiply back. Always check the parts add to x.
  • Pie-chart angle = \dfrac{\text{part}}{\text{total}}\times 360°. Simplify first; angles must total 360°.
  • Direct vs inverse: “Double the first, does the second double or halve?” Double \to direct (keep \frac{x}{y}); halve \to inverse (keep xy).
  • Inverse shortcut: x_1 y_1 = x_2 y_2. Speed \times time, workers \times days and pumps \times hours each stay constant.

Spot the Mistake

Common Exam Mistakes

  • Treating every word problem as direct. “30 pens cost ₹150, find 24 pens” is direct (fewer pens, less money). But “6 taps fill a tank in 20 min, find 8 taps” is inverse. Always ask which way the second quantity moves.
  • Forgetting the units in a map scale. The RF’s second term is in centimetres. To reach kilometres you must divide by 1{,}00{,}000, not by 1000.
  • Sharing without adding the terms. To split in 5:3:2, one part is \dfrac{x}{5+3+2}, not \dfrac{x}{5}. The denominator is the sum of all terms.
  • Pie slices that miss 360°. If your angles add to anything but 360°, a part is wrong, recompute before drawing.
  • Mixing up the inverse equation. Inverse proportion uses x_1 y_1 = x_2 y_2 (a product), not \dfrac{x_1}{x_2}=\dfrac{y_1}{y_2} (which is direct).

Share It Out!

A quick, friendly tool: enter a total and a ratio, and watch the whole split into its parts, perfect for prize money, paint mixes or rehearsal time.

Ratio splitter
Enter a total and a ratio like 5:3:2, then press Split.

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. On a map with RF 1 : 25{,}00{,}000, 1 cm represents ______ km on the ground.
  2. When 360° is shared in the ratio 3 : 2 : 1, the largest slice measures ______.
  3. If 4 taps fill a tank in 90 minutes, then 6 identical taps fill it in ______ minutes.

Short answer (2 marks each)

  1. Two towns are 8 cm apart on a map of scale 1 : 60{,}00{,}000. Find the ground distance in kilometres.
  2. Share ₹4500 among three workers in the ratio 4 : 3 : 2. How much does each receive?

Long answer (3 marks each)

  1. A family’s monthly budget is Food ₹9000, Rent ₹6000, Transport ₹3000, Savings ₹6000. Find the pie-chart angle for each item and check they total 360°.
  2. 15 workers build a wall in 28 days. Working at the same rate, how many days would 21 workers take? State whether the proportion is direct or inverse.

HOTS (Higher Order Thinking)

  1. In a pie chart of 720 students, the “Sports” slice measures 100°. How many students chose Sports?
  2. Two pipes can fill a tank in 12 hours and 18 hours respectively. If both run together, how long will the tank take to fill?

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): In a pie chart, a category making up 40\% of the data is drawn with a 144° slice.   Reason (R): A slice’s angle equals its fraction of the whole multiplied by 360°.
  1. 25{,}00{,}000 cm \div 1{,}00{,}000 = \mathbf{25} km.
  2. The terms add to 3+2+1 = 6, so one part = 360° \div 6 = 60°. The largest slice is 3 \times 60° = \mathbf{180°}.
  3. More taps, less time, inverse. 4 \times 90 = 6 \times x, so x = \dfrac{360}{6} = \mathbf{60} minutes.
  4. 1 cm = 60{,}00{,}000 cm = 60 km, so 8 cm = 8 \times 60 = \mathbf{480} km.
  5. Terms add to 4+3+2 = 9, so one part = 4500 \div 9 = ₹500. Shares: 4\times500 = \mathbf{₹2000}, 3\times500 = \mathbf{₹1500}, 2\times500 = \mathbf{₹1000} (check: 2000+1500+1000 = 4500 ✓).
  6. Total = 9000+6000+3000+6000 = ₹24{,}000. Each angle = \dfrac{\text{part}}{24000}\times 360°: Food = \dfrac{9000}{24000}\times360° = 135°, Rent = 90°, Transport = 45°, Savings = 90° (135+90+45+90 = 360° ✓).
  7. More workers, fewer days, inverse. 15 \times 28 = 21 \times x, so x = \dfrac{420}{21} = \mathbf{20} days.
  8. Each degree stands for \dfrac{720}{360} = 2 students, so 100° represents 100 \times 2 = \mathbf{200} students.
  9. In one hour the pipes fill \dfrac{1}{12} and \dfrac{1}{18} of the tank. Together: \dfrac{1}{12}+\dfrac{1}{18} = \dfrac{3}{36}+\dfrac{2}{36} = \dfrac{5}{36} per hour. The full tank takes \dfrac{36}{5} = \mathbf{7.2} hours = 7 hours 12 minutes.
  10. (a), Both are true and R is the correct explanation: 40\% \times 360° = 0.4 \times 360° = 144°, exactly the rule R states.

Connections to Other Chapters

How This Chapter Links Across the Book

  • Percentages and comparing quantities: a percentage is just a ratio out of 100, and pie-chart angles are percentages turned into slices of 360°.
  • Fractions: sharing in a ratio and adding work rates (\frac{1}{2}+\frac{1}{3}) both rest on confident fraction arithmetic.
  • Geometry (angles and the circle): building a pie chart uses a protractor and the fact that angles around a centre sum to 360°.
  • Algebra and graphs: direct proportion y = kx gives a straight line through the origin, while inverse proportion xy = k traces a curve, ideas you will meet again with linear equations and graphs.

Glossary

Key Terms

  • Proportion: a statement that two ratios are equal, e.g. 12:4 :: 9:3.
  • Representative Fraction (RF): the ratio of a map distance to the matching ground distance, e.g. 1 : 50{,}00{,}000.
  • Scale: how much real distance one unit on a map stands for; a large-scale map shows a small area in detail, a small-scale map a large area with little detail.
  • Multi-term ratio: a ratio comparing three or more quantities, such as 6:4:2:1.
  • Sharing in a ratio: splitting a whole so the parts are in a given ratio, using each part = x\times\frac{\text{term}}{\text{sum of terms}}.
  • Pie chart: a circular graph that shows each part as a slice whose angle is proportional to its share of 360°.
  • Direct proportion: two quantities with a constant ratio (\frac{x}{y}=k); they rise and fall together.
  • Inverse proportion: two quantities with a constant product (xy=k); one rises as the other falls.