The difference between a number and its three-fifths is 50, What is the number ?
(A) 75
(B) 100
(C) 125
(D) None of these
Solution:
Let the number be x Then, $$\eqalign{ & \Leftrightarrow x - \frac{3}{5}x = 50 \cr & \Leftrightarrow \frac{2}{5}x = 50 \cr & \Leftrightarrow x = \left( {\frac{{50 \times 5}}{2}} \right) \cr & \Leftrightarrow x = 125 \cr} $$
3.
The sum of five consecutive odd numbers is 575. What is the sum of the next set of five consecutive odd numbers ?
The sum of two number is 25 and their difference is 13. Find their product.
(A) 104
(B) 114
(C) 315
(D) 325
Solution:
Let the numbers be x and y Then, x + y = 25 and x - y = 13 4xy = (x + y)2 - (x- y)2 = (25)2 - (13)2 = (625 - 169) = 456 ∴ xy = 114
5.
There are two numbers such that the sum of twice the first number and thrice the second number is 100 and the sum of thrice the first number and twice the second number is 120. Which is the larger number ?
(A) 12
(B) 14
(C) 32
(D) 35
Solution:
Let the numbers be x and y Then, 2x + 3y = 100 ..... (i) And, 3x + 2y = 120 ..... (ii) Adding (i) and (ii), we get : 5x + 5y = 220 or x + y = 44 ..... (iii) Subtracting (i) from (ii): we get : x - y = 20 ..... (iv) Adding (iii) and (iv), we get : 2x = 64 or x = 32 Putting x = 32 in (iii), we get : y = 12 Hence, larger number = 32
6.
The sum of the squares of two numbers is 3341 and the difference of their squares is 891. The numbers are :
(A) 25, 36
(B) 25, 46
(C) 35, 46
(D) None of these
Solution:
Let the numbers be x and y. Then, x2 + y2 = 3341..... (i) And, x2 - y2 = 891..... (ii) Adding (i) and (ii), we get : 2x2 = 4232 or x2 = 2116 or x = 46 Subtracting (ii) from (i), we get : 2y2 = 2450 or y2 = 1225 or y = 35 So, the numbers are 35 and 46
7.
The sum of two numbers is 75 and their difference is 25. The product of the two numbers is :
(A) 1350
(B) 1250
(C) 125
(D) 1000
Solution:
Let the numbers be a and b According to the question, $$\eqalign{ & a + b = 75 \cr & a - b = 25 \cr & \because {\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} = 4ab \cr & \Rightarrow {75^2} - {25^2} = 4ab \cr & \Rightarrow 4ab = \left( {75 + 25} \right)\left( {75 - 25} \right) \cr & \left[ {\because {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)} \right] \cr & \Rightarrow 4ab = 100 \times 50 \cr & \Rightarrow ab = \frac{{100 \times 50}}{4} \cr & \Rightarrow ab = 1250 \cr} $$
8.
If a number is added to two-fifths of itself, the value so obtained is 455. What is the number ?
(A) 325
(B) 350
(C) 400
(D) 420
Solution:
Let the number be x Then, $$\eqalign{ & \Leftrightarrow x + \frac{2}{5}x = 455 \cr & \Leftrightarrow \frac{7}{5}x = 455 \cr & \Leftrightarrow x = \left( {\frac{{455 \times 5}}{7}} \right) \cr & \Leftrightarrow x = 325 \cr} $$
9.
If a number is multiplied by two-third of itself the value so obtained is 864. What is the number ?
(A) 34
(B) 36
(C) 38
(D) 44
Solution:
Let the number be x Then, $$\eqalign{ & \Leftrightarrow x \times \frac{2}{3}x = 864 \cr & \Leftrightarrow \frac{2}{3}{x^2} = 864 \cr & \Leftrightarrow {x^2} = \left( {\frac{{864 \times 3}}{2}} \right) \cr & \Leftrightarrow {x^2} = 1296 \cr & \Leftrightarrow x = \sqrt {1296} \cr & \Leftrightarrow x = 36 \cr} $$
10.
If 50 is subtracted from two-third of number, the result is equal to sum of 40 and one-fourth of that number. What is the number ?
(A) 174
(B) 216
(C) 246
(D) 336
Solution:
Let the number be x Then, $$\eqalign{ & \Leftrightarrow \frac{2}{3}x - 50 = \frac{1}{4}x + 40 \cr & \Leftrightarrow \frac{2}{3}x - \frac{1}{4}x = 90 \cr & \Leftrightarrow \frac{{5x}}{{12}} = 90 \cr & \Leftrightarrow x = \left( {\frac{{90 \times 12}}{5}} \right) \cr & \Leftrightarrow x = 216 \cr} $$