Problems On Numbers - Study Mode

[#36] The difference between the numerator and the denominator of a fraction is 5. If 5 is added to its denominator, the fraction is decreased by $$1frac{1}{4}$$. Find the value of the fraction.
Correct Answer

(B) $$2frac{1}{4}$$

Explanation

Solution: Let the denominator be x Then, numerator = x + 5 Now, $$eqalign{
& Leftrightarrow frac{{x + 5}}{x} - frac{{x + 5}}{{x + 5}} = frac{5}{4} cr
& Leftrightarrow frac{{x + 5}}{x} = frac{5}{4} + 1 cr
& Leftrightarrow frac{{x + 5}}{x} = frac{9}{4} cr
& Leftrightarrow frac{{x + 5}}{x} = 2frac{1}{4} cr} $$ So, the fraction is $$2frac{1}{4}$$

[#37] A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is :
Correct Answer

(D) 280

Explanation

Solution: Let the number be x Then, $$eqalign{
& Leftrightarrow left( {frac{1}{5}x + 4}
ight) = left( {frac{1}{4}x - 10}
ight) cr
& Leftrightarrow frac{x}{{20}} = 14 cr
& Leftrightarrow x = 14 imes 20 cr
& Leftrightarrow x = 280 cr} $$

[#38] The ratio between a two-digit number and the sum of the digits of that number is 4 : 1. If the digit in the unit's place is 3 more than the digit in the ten's place, then the number is ?
Correct Answer

(B) 36

Explanation

Solution: Let the ten's digit be x Then, units digit = x + 3 Number = 10x + (x + 3) = 11x + 3 Sum of digits = x + (x + 3) = 2x + 3 $$eqalign{
& herefore frac{{11x + 3}}{{2x + 3}} = frac{4}{1} cr
& Leftrightarrow 11x + 3 = 8x + 12 cr
& Leftrightarrow 3x = 9 cr
& Leftrightarrow x = 3 cr} $$ Hence, Required number = 11x + 3 = 11 × 3 + 3 = 36

[#39] The difference between two positive integers is 3. If the sum of their squares is 369, then the sum of the numbers is :
Correct Answer

(B) 27

Explanation

Solution: Let the numbers be x and (x + 3) Then, $$eqalign{
& Leftrightarrow {x^2} + {left( {x + 3}
ight)^2} = 369 cr
& Leftrightarrow {x^2} + {x^2} + 9 + 6x = 369 cr
& Leftrightarrow 2{x^2} + 6x - 360 = 0 cr
& Leftrightarrow {x^2} + 3x - 180 = 0 cr
& Leftrightarrow left( {x + 15}
ight)left( {x - 12}
ight) = 0 cr
& Leftrightarrow x = 12 cr} $$ So, the numbers are 12 and 15 ∴ Required sum = (12 + 15) = 27

[#40] A number consists of two digits. If the digits interchange place and the new number is added to the original number, then the resulting number will be divisible by :
Correct Answer

(D) 11

Explanation

Solution: Let the ten's digit be x and unit's digit be y Then, number = 10x + y Number obtained by interchanging the digits = 10y + x ∴ (10x + y) + (10y + x) = 11(x + y), which is divisible by 11