Exercise 10


1. In a two-digit number, the digit at unit place is 1 more than twice of the digit at tens place. If the digit at unit and tens place be interchanged, then the difference between the new number and original number is less than 1 to that of original number. What is the original number?
(a) 52
(b) 73
(c) 25
(d) 49
(e) 37
Ans.e

Let the original number be l0 x + y

y = 2x + 1....(i)

and (l0y + x) – (10x + y) = 10x + y – 1

or, 9y – 9x = l0x + y – 1

or, l9x – 8y = 1 ...(ii)

Putting the value of (i) in equation (ii) we get,

19x – 8(2x + 1) = 1

or, 19x – 16x – 8 = 1

or, 3x = 9 or, x = 3

So, y = 2 × 3 + 1 = 7

\ original number = 10 × 3 + 7 = 37

2. Free notebooks were distributed equally among children of a class. The number of notebooks each child got was one-eighth of the number of children. Had the number of children been half, each child would have got 16 notebooks. How many notebooks were distributed in all?
(a) 432
(b) 640
(c) 256
(d) 512
(e) None of these
Ans.d

In case I: Let the no. of children = x.

Hence, total no. of notebooks distributed

18x.x or x28.......(i)

In case II: no.of children = x2

Now, the total no. of notebooks

= 16 × x2 .......(ii)

Comparing (i) & (ii), we get

x28 = 8x

or, x = 64

Hence, total no. of notebooks

= \(\frac{64 \times 64}{8} = 512\)

3. Twenty times a positive integer is less than its square by 96. What is the integer?
(a) 24
(b) 20
(c) 30
(d) Cannot be determined
(e) None of these
Ans.a

Let the positive integer be x.

Now, x2 – 20x = 96

or, x2 – 20x – 96 = 0

or, x2 – 24x + 4x – 96 = 0

or, x(x – 24) + 4(x – 24) = 0

or, (x – 24)(x + 4) = 0

or, x = 24, – 4

x ≠ –4 because x is a positive integer

4. A man starts going for morning walk every day. The distance walked by him on the first day was 2 km. Everyday he walks half of the distance walked on the previous day. What can be the maximum total distance walked by him in his lifetime?
(a) 4 km
(b) 20 km
(c) 8 km
(d) Data inadequate
(e) None of these
Ans.a

Distance walked by man

= 2 + 1 + 12 + 14 + 18 + 116....∞

The above series is in infinity GP.

\(s_{\infty } = \frac{2}{1 - \frac{1}{2}} = 4 \; km\)

Note: If the series is in GP then

\(\frac{second \;\; term}{first \;\; term} = \frac{third \;\; term}{second \;\; term} ....= Common \;\; Ratio\)

Sum of the Infinity GP = \(\frac{first \;\; term}{\left | 1 - common \;\; ratio \right |}\)

5. The digit in the units place of a number is equal to the digit in the tens place of half of that number and the digit in the tens place of that number is less than the digit in units place of half of the number by 1. If the sum of the digits of the number is seven, then what is the number?
(a) 52
(b) 16
(c) 34
(d) Data inadequate
(e) None of these
Ans.a

Let ½ of the no. = 10x + y

and the no. = 10V + W From the given conditions,

W = x and V = y – 1

Thus the no. = 10(y – 1) + x…..(A)

∴ 2(10x + y) = 10(y – 1) + x ⇒ 8y – 19x = 10…(i)

Again, from the question,

V + W = 7 ⇒ y – 1 + x = 7

∴ x + y = 8 …(ii)

Solving equations (i) and (ii), we get x = 2 and y = 6.

∴ From equation (A), Number = 10(y – 1) + x = 52

6. The difference between a two-digit number and the number obtained by interchanging the digits is 9. What is the difference between the two digits of the number?
(a) 8
(b) 2
(c) 7
(d) Cannot be determined
(e) None of these
Ans.e

Suppose the two-digit number be 10x + y.

Then we have been given

l0x + y – (10y + x) = 9

⇒ 9x – 9y = 9

⇒ x – y = 1

Hence, the required difference = 1

Note that if the difference between a two-digit number and the number obtained by interchanging the digits is D, then the difference between the two digits of the number = D9

7. The difference between a number and its three-fifths is 50.What is the number?
(a) 75
(b) 100
(c) 125
(d) Cannot be determined
(e) None of these
Ans.c

Suppose the number is N.

Then N - 35N = 50

2N5 = 50

\(N = \frac{50 \times 5}{2} = 125\)

8. If the numerator of a fraction is increased by 2 and the denominator is increased by 1, the fraction becomes 58 and if the numerator of the same fraction is increased by 3 and the denominator is increased by I the fraction becomes 34. What is the original fraction?
(a) Data inadequate
(b) 27
(c) 47
(d) 37
(e) None of these
Ans.d

Let the original fraction be xy.

Then \(\frac{x + 2}{y + 1} = \frac{5}{8}\) or, 8x – 5y = – 11........ (i)

Again, \(\frac{x + 3}{y + 1} = \frac{3}{4}\) or, 4x – 3y = –9........ (ii)

Solving, (i) and (ii) we get x = 3 and y = 7

∴ fraction = 37

9. If 2x + 3y = 26; 2y + z = 19 and x + 2z = 29, what is the value of x + y + z ?
(a) 18
(b) 32
(c) 26
(d) 22
(e) None of these
Ans.d

On solving equation we get

x = 7, y = 4, z = 11

10. If the sum of a number and its square is 182, what is the number?
(a) 15
(b) 26
(c) 28
(d) 91
(e) None of these
Ans.e

Let the number = x

Then, x2 + x = 182

or, x2 + x – 182 = 0

or, x + 14x – 13x – 182 = 0

or, x(x + 14) – 13(x + 14) = 0

or, (x – 13)(x + 14) = 0

or, x = 13 (negative value is neglected)