Class 9 Mathematics

Chapter 3 Solutions — The World of Numbers

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Solutions

Overview

Step-by-step NCERT solutions for The World of Numbers (Chapter 3, NCERT Class 9 Mathematics) — the full working for every question, not just the final answer. You can also read the The World of Numbers textbook chapter.

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What these solutions cover

All 43 questions in The World of Numbers are solved in the PDF. Here's what's inside, exercise by exercise:

Natural Numbers and Early Counting Systems

A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?
Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.
We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
*Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?

Integers and Brahmagupta's Laws

The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature?
A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Write this sequence as an equation using integers and calculate his final financial standing.
3.
Calculate the following using Brahmagupta's laws:
- (i) (−12) × 5
- (ii) (−8) × (−7)
- (iii) 0 − (−14)
- (iv) (−20) ÷ 4
Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., 10 − (−5) = 15).

Fractions, Rational Numbers, and Their Arithmetic

1.
Prove that the following rational numbers are equal:
- (i) 2/3 and 4/6
- (ii) 5/4 and 10/8
- (iii) −3/5 and −6/10
- (iv) 9/3 and 3
2.
Find the sum:
- (i) 2/5 + 3/10
- (ii) 7/12 + 5/8
- (iii) −4/7 + 3/14
3.
Find the difference:
- (i) 5/6 − 1/4
- (ii) 11/8 − 3/4
- (iii) −7/9 − (−2/3)
4.
Find the product:
- (i) 2/3 × 3/10
- (ii) 7/11 × 5/8
- (iii) −4/7 × 5/14
5.
Find the quotient:
- (i) 2/3 ÷ 3/10
- (ii) 7/11 ÷ 5/8
- (iii) −4/7 ÷ 5/14
Show that: (1/2 + 3/4) × 8/3 = 1/2 × 8/3 + 3/4 × 8/3.
Simplify the following using the distributive property: 7/9 × (6/7 − 3/4).
Find the rational number x such that: 5/6 (x + 3/5) = 5/6 · x + 1/2.

Representation of Rational Numbers on the Number Line

Represent the rational numbers 2/3, −5/4 and 1(1/2) on a single number line.
Find three distinct rational numbers that lie strictly between −1/2 and 1/4.
Simplify the expression: (−1/4) + (5/12).
A tailor has 15(3/4) metres of fine silk. If making one kurta requires 2(1/4) metres of silk, exactly how many kurtas can he make?
Find three rational numbers between 3.1415 and 3.1416.
*Can you think of other way(s) to find a rational number between any two rational numbers?

Decimal Expansions, Cyclic Numbers, and Irrational Numbers

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 7/20, 4/15 and 13/250. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.
Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 2/13? Now compute 3/13, 4/13, etc. What do you notice?
3.
Classify the following numbers as rational or irrational:
- (i) √81
- (ii) √12
- (iii) 0.33333…
- (iv) 0.123451234512345…
- (v) 1.01001000100001…
- (vi) 23.560185612239874790120. Find the explicit fractions in case they are rational.
The number 0.(9) (which means 0.99999…) is a rational number. Using algebra (let x = 0.(9), multiply by 10, and subtract), explain why 0.(9) is exactly equal to 1.
*We have seen that the repeating block of 1/7 is a cyclic number. Try to find more numbers (n) whose reciprocals (1/n) produce decimals with repeating blocks that are cyclic.

Exercises

1.
Convert the following rational numbers into a terminating decimal or a non-terminating and repeating decimal, whichever the case may be, by the process of long division:
- (i) 3/50
- (ii) 2/9
Prove that √5 is an irrational number.
3.
Convert the following decimal numbers into the form p/q:
- (i) 12.6
- (ii) 0.0120
- (iii) 3.052
- (iv) 1.235
- (v) 0.23
- (vi) 2.05
- (vii) 2.125
- (viii) 3.125
- (ix) 2.1625
4.
Locate the following rational numbers on the number line:
- (i) 0.532
- (ii) 1.15
Find 6 rational numbers between 3 and 4.
Find 5 rational numbers between 2/5 and 3/5.
Find 5 rational numbers between 1/6 and 2/5.
If x/3 + x/5 = 16/15, find the rational number x.
Let a and b be two non-zero rational numbers such that a + 1/b = 0. Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer.
A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form p/10⁴, where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by 2⁴ or 5⁴? Give reasons.
Without performing division, determine whether the decimal expansion of 18/125 is terminating or non-terminating. If it terminates, state the number of decimal places.
A rational number in its lowest form has denominator 2³ × 5. How many decimal places will its decimal expansion have? Explain your answer.
Let a = 7/12 and b = 5/6. Express both a and b in the form k₁/m and k₂/m where k₁, k₂ and m are integers and k₂ – k₁ > 6. Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition k₂ – k₁ > n + 1 is necessary to find n such rational numbers between the two rational numbers a and b using this method.
Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero.
Show that the rational number (a + b)/2 lies between the rational numbers a and b.
Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14, which is referred to as the square root spiral.

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