Exercise 10.1 — Direct Proportion

Introduction of direct proportion and its applications.

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Exercise 10.1 – Direct Proportion: Solved Word Problems & Unitary Method | Class 8 Maths | EduBadi

Exercise 10.1 – Direct Proportion: Solved Word Problems Using the Unitary Method

Exercise 10.1 from Chapter 10, "Direct and Inverse Proportions," of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) focuses entirely on direct proportion — situations where two quantities increase or decrease together at the same rate. From the cost of cloth and apples to the speed of trains and the design of microchips, this exercise shows how a single proportionality formula can solve a huge variety of real-life problems.

Direct proportion problems are extremely common in CBSE, Telangana, and Andhra Pradesh board exams, and the technique learned here — setting up a table and applying the proportion formula — is also the foundation for the ratio and proportion concepts used throughout Class 8 and 9.

What Is Direct Proportion?

Two quantities x and y are said to be in direct proportion if they increase or decrease together, and the ratio x/y always stays the same (a constant, usually called k). This relationship is written as x ∝ y, read as "x is directly proportional to y."

If x ∝ y, then x/y = k (a constant)

If x₁ and y₁ are one pair of corresponding values, and x₂ and y₂ are another pair of corresponding values for the same two quantities, the direct proportion relationship gives us a very useful equation:

x₁/y₁ = x₂/y₂

💡 Key Insight: This single equation is the basis for solving every question in this exercise. The general method is: identify the two related quantities, write down the known pair (x₁, y₁) and the pair with one unknown value (x₂, y₂), then cross-multiply to find the unknown.

Exercise 10.1 – Solved Questions, Step by Step

Question 1

Cost of Cloth for Different Lengths

Basic Direct Proportion

Given that 5 metres of cloth costs ₹210, as the length of cloth increases, the cost increases too — so length and cost are in direct proportion. For each part, the formula x₁/y₁ = x₂/y₂ is applied with x₁ = 5 and y₁ = 210.

Length of Cloth (m)	Calculation	Cost (₹)
(i) 2 m	y₂ = 210 × (2/5) = 42 × 2	84
(ii) 4 m	y₂ = 210 × (4/5) = 42 × 4	168
(iii) 10 m	y₂ = 210 × (10/5) = 42 × 10	420
(iv) 13 m	y₂ = 210 × (13/5) = 42 × 13	546

📐 Simplify First: Notice that 210/5 = 42 in every part. Once you find this constant (the cost per metre), you can simply multiply it by any new length — this is the essence of the unitary method.

Question 2

Completing a Table — Cost of Apples

Direct Proportion Table

Given that 1 apple costs ₹8, the table below is completed by applying x₁/y₁ = x₂/y₂ with x₁ = 1 and y₁ = 8 for each new number of apples. Since x₁ = 1, this simplifies to y₂ = 8 × (number of apples) — a direct application of the unitary method.

Number of Apples	Calculation	Cost (₹)
1	Given	8
(i) 4	8 × 4	32
(ii) 7	8 × 7	56
(iii) 12	8 × 12	96
(iv) 20	8 × 20	160

Question 3

Cost of Paddy Bags

Decreasing Direct Proportion

Given that 48 bags of paddy cost ₹16,800, find the cost of 36 bags. As the number of bags decreases, the cost decreases too — so this is still a direct proportion, just with smaller values.

48/36 = 16800/y₂ ⇒ y₂ = 16800 × (36/48) = 4200 × 3 = 12600

Cost of 36 bags = ₹12,600

📐 Direct Proportion Doesn't Always Mean "Increasing": Direct proportion simply means both quantities move in the same direction — both can decrease together, as in this question, and the relationship still holds.

Question 4

Family Expenditure Based on Number of Members

Direct Proportion — Average Expenditure

A family of 4 members has a monthly average expenditure of ₹2,800. Find the average expenditure for a family of 3 members. As the number of members decreases, the total/average expenditure decreases too.

4/3 = 2800/y₂ ⇒ y₂ = 2800 × (3/4) = 700 × 3 = 2100

Average monthly expenditure for 3 members = ₹2,100

Question 5

Scale Model of a Ship's Mast

Direct Proportion — Scale Models

A ship of length 28 m has a mast of height 12 m. In a model of the ship, the mast height is 9 cm. Find the model's length. Since the model is a proportional scaled-down version, the ship's length and mast height are in direct proportion with the model's length and mast height.

28/x₂ = 12/9 ⇒ x₂ = 28 × (9/12) = 7 × 3 = 21

Length of the model ship = 21 cm

💡 Key Insight: This question shows direct proportion in action for scale models — every dimension of a model shrinks or grows by the same ratio as every other dimension, compared to the real object.

Question 6

Pole Heights and Shadow Lengths

Direct Proportion — Shadows

A vertical pole of height 5.6 m casts a shadow 3.2 m long. At the same time (same sun angle), the height of any pole and the length of its shadow are in direct proportion.

(i) Shadow Cast by a 10.5 m Pole

5.6/10.5 = 3.2/y₂ ⇒ y₂ = 3.2 × (10.5/5.6) = 6

Shadow length for a 10.5 m pole = 6 m

(ii) Height of a Pole with a 5 m Shadow

5.6/x₂ = 3.2/5 ⇒ x₂ = 5.6 × (5/3.2) = 8.75

Height of pole with a 5 m shadow = 8.75 m

📐 Why Does This Work? At any given moment, sunlight falls at the same angle everywhere nearby, so every object's height-to-shadow ratio stays constant. This is the same principle used in surveying to measure the height of tall objects (like trees or buildings) using shadows.

Question 7

Distance Travelled by a Truck

Direct Proportion — Speed, Time & Distance

A truck travels 14 km in 25 minutes. At the same speed, how far can it travel in 5 hours? First, convert 5 hours into minutes: 5 × 60 = 300 minutes. Since speed is constant, time and distance are in direct proportion.

14/x₂ = 25/300 ⇒ x₂ = 14 × (300/25) = 14 × 12 = 168

Distance travelled in 5 hours = 168 km

📐 Unit Conversion Comes First: Before setting up the proportion, both time values must be in the same unit (minutes, in this case). Converting 5 hours to 300 minutes is the crucial first step — skipping it would give a completely wrong ratio.

Question 8

Number of Paper Sheets for a Given Weight

Direct Proportion — Mixed Fractions & Unit Conversion

If 12 sheets of paper weigh 40 grams, how many sheets would weigh 16⅔ kg? First, convert the mixed fraction and the unit to grams:

16⅔ kg = 50/3 kg = (50/3) × 1000 g = 50000/3 grams

Now apply the direct proportion formula, with the number of sheets (x) and weight in grams (y):

12/x₂ = 40/(50000/3) ⇒ x₂ = (12/40) × (50000/3) = 5000

Number of sheets weighing 16⅔ kg = 5000 sheets

💡 Key Insight: Mixed fractions like 16⅔ should always be converted to a single fraction (50/3) before use in calculations — this avoids errors when multiplying or dividing. Also, converting kilograms to grams (multiplying by 1000) ensures both weights in the proportion are in the same unit.

Question 9

Train Speed — Distance and Time Calculations

Direct Proportion — Two-Part Question

A train moves at a constant speed of 75 km/hr.

(i) Distance Travelled in 20 Minutes

Since the train covers 75 km in 1 hour (60 minutes), and time & distance are in direct proportion at constant speed:

60/20 = 75/y₂ ⇒ y₂ = 75 × (20/60) = 25

Distance in 20 minutes = 25 km

(ii) Time Required to Cover 250 km

1/x₂ = 75/250 ⇒ x₂ = 250/75 = 10/3 = 3⅓

Time required to cover 250 km = 3⅓ hours

Question 10

Actual Length of a Microchip from Its Design Scale

Direct Proportion — Scale Ratio (40:1)

A microchip design uses a scale of 40:1, meaning the design is 40 times larger than the actual chip. If the design's length is 18 cm, find the actual chip length.

40/18 = 1/y₂ ⇒ y₂ = 18/40 = 9/20

Actual length of the microchip = 9/20 cm = 0.45 cm

📐 Scale Ratios as Proportions: A scale of "40:1" simply means the design-to-actual ratio is 40/1 — this can be treated as one pair (x₁, y₁) = (40, 1) in the proportion formula, just like any other pair of corresponding values.

Question 11

Ratio of Doctors to Lawyers from Average Ages

Algebraic Approach — Forming & Solving Equations

The combined average age of doctors and lawyers is 40. The doctors' average age is 35, and the lawyers' average age is 50. Find the ratio of the number of doctors to the number of lawyers. This question is different from the rest — it requires setting up an algebraic equation rather than a direct proportion table.

Step 1: Let the number of doctors = x and the number of lawyers = y.
Step 2 — Total age (combined average): Sum of all ages = average × total people = 40(x + y) = 40x + 40y
Step 3 — Total age (separately): Sum of doctors' ages = 35x, and sum of lawyers' ages = 50y. So total = 35x + 50y
Step 4 — Equate the two expressions for total age: 40x + 40y = 35x + 50y
Step 5 — Simplify: 40x − 35x = 50y − 40y ⇒ 5x = 10y
Step 6 — Find the ratio: x/y = 10/5 = 2/1

Ratio of doctors to lawyers = 2 : 1

💡 Key Insight: The "combined average" of two groups is a weighted average — it depends not just on the individual averages (35 and 50) but also on how many people are in each group. Here, since the combined average (40) is closer to the doctors' average (35) than to the lawyers' average (50), it makes sense that there are more doctors than lawyers — and indeed, the ratio 2:1 confirms there are twice as many doctors as lawyers.

Common Mistakes to Avoid in Exercise 10.1

Forgetting unit conversions: Questions involving hours/minutes (Q7, Q9), kilograms/grams (Q8), or metres/centimetres (Q5) all require converting to the same unit before setting up the proportion. This is the single most common source of errors in this exercise.
Mixing up which value goes where: Always keep x₁/y₁ = x₂/y₂ consistent — if x represents the "first" quantity (like length or time) and y represents the "second" (like cost or distance), don't accidentally swap them partway through.
Not simplifying mixed fractions first: A value like 16⅔ must be converted to an improper fraction (50/3) before being used in any multiplication or division (Q8).
Assuming direct proportion always means "increasing": As Q3 shows, direct proportion also applies when both quantities decrease together — the defining feature is a constant ratio, not a particular direction of change.
Treating average-based problems as simple proportion: Q11 cannot be solved with the basic x₁/y₁ = x₂/y₂ formula — it requires setting up and solving an algebraic equation based on how averages combine.

What Exercise 10.1 Prepares You For

Having mastered direct proportion here, the next step in this chapter is inverse proportion — situations where one quantity increases as the other decreases (for example, more workers completing a job in less time). The same "set up a table, then cross-multiply" approach is used, but with a different relationship between the quantities.

These proportional reasoning skills are also essential for Comparing Quantities Using Proportion, where ratios, percentages, profit and loss, and simple/compound interest all build on the same core idea of a constant ratio between two related quantities. In Class 9 and Class 10, these concepts extend further into similar triangles (where corresponding sides are in proportion) and into algebraic and graphical representations of proportional relationships.