15. The difference between two numbers is 1365. When the large number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is :
Solution:
Let the numbers be x and (x + 1365)
Then,
⇒ x + 1365 = 6x + 15
⇒ 5x = 1350
⇒ x = 270
16. What is the greater of the two numbers whose product is 1092 and the sum of the two numbers exceeds their difference by 42 ?
Solution:
Let the numbers be x and y
Then,
$$xy = 1092.....(i)$$
And,
$$\eqalign{
\Leftrightarrow \left( {x + y} \right) - \left( {x - y} \right) = 42 \cr
\Leftrightarrow 2y = 42 \cr
\Leftrightarrow y = 21 \cr} $$
Putting y = 21 in (i), we get :
$$x = \frac{{1092}}{{21}} = 52$$
Hence, greater number = 52
17. 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 ?
Solution:
Let the number be x
Then,
$$\eqalign{
\Leftrightarrow \frac{2}{3}x - 50 = \frac{1}{4}x + 40 \cr
\Leftrightarrow \frac{2}{3}x - \frac{1}{4}x = 90 \cr
\Leftrightarrow \frac{{5x}}{{12}} = 90 \cr
\Leftrightarrow x = \left( {\frac{{90 \times 12}}{5}} \right) \cr
\Leftrightarrow x = 216 \cr} $$
18. The sum of two numbers is 40 and their difference is 4. The ratio of the numbers is :
Solution:
Let the numbers be x and y
Then,
$$\eqalign{
\Leftrightarrow \frac{{x + y}}{{x - y}} = \frac{{40}}{4} \cr
\Leftrightarrow \frac{{x + y}}{{x - y}} = 10 \cr
\Leftrightarrow \left( {x + y} \right) = 10\left( {x - y} \right) \cr
\Leftrightarrow 9x = 11y \cr
\Leftrightarrow \frac{x}{y} = \frac{{11}}{9} \cr
\Leftrightarrow x:y = 11:9 \cr} $$
19. The sum of seven consecutive numbers is 175. What is the difference between twice the largest number and thrice the smallest number ?
Solution:
Let the seven numbers be x, (x + 1), (x + 2), (x + 3), (x + 4), (x + 5) and (x + 6)
Then,
⇔ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6) = 175
⇔ 7x + 21 = 175
⇔ 7x = 154
⇔ x = 22
Required difference :
= 2(x + 6) - 3x
= 12 - x
= 12 - 22
= -10
20. The sum of four numbers is 64. If you add 3 to the first number, 3 is subtracted from the second number, the third is multiplied by 3 and the fourth is divided by 3, then all the results are equal. What is the difference between the largest and the smallest of the original numbers ?
Solution:
Let the four numbers be , A, B, C and D
Let A + 3 = B - 3 = 3C = $$\frac{D}{3}$$ = x
Then,
A = x - 3
B = x + 3
C = $$\frac{x}{3}$$
D = 3x
$$\eqalign{
\Leftrightarrow A + B + C + D = 64 \cr
\Leftrightarrow \left( {x - 3} \right) + \left( {x + 3} \right) + \frac{x}{3} + 3x = 64 \cr
\Leftrightarrow 5x + \frac{x}{3} = 64 \cr
\Leftrightarrow 16x = 192 \cr
\Leftrightarrow x = 12 \cr} $$
Thus, the numbers are 9, 15, 4 and 36
∴ Required difference :
= (36 - 4)
= 32
21. If the square of a two-digit number is reduced by the square of the number formed by reversing the digits of the number, the final result is :
Solution:
Let the two-digit number be 10x + y
Then, number formed by reversing the digits = 10y + x
Difference of square of the numbers :
$$ = {\left( {10x + y} \right)^2} - {\left( {10y + x} \right)^2}$$
$$ = \left( {100{x^2} + {y^2} + 20xy} \right) - $$ $$\left( {100{y^2} + {x^2} + 20xy} \right)$$
$$ = 99\left( {{x^2} - {y^2}} \right)$$ which is divisible by both 9 and 11
