Problems On Numbers - Study Mode

[#61] If the numerator of a fraction is increased by 200% and the denominator is increased by 300%, the resultant fraction is $$frac{15}{26}$$. What was the original fraction ?
Correct Answer

(D) $$frac{10}{13}$$

Explanation

Solution: Let the fraction be $$frac{x}{y}$$ Then, $$eqalign{
& Leftrightarrow frac{{x + 200\% { ext{ of }}x}}{{y + 300\% { ext{ of }}y}} = frac{{15}}{{26}} cr
& Leftrightarrow frac{{3x}}{{4y}} = frac{{15}}{{26}} cr
& Leftrightarrow frac{x}{y} = frac{{15}}{{26}} imes frac{4}{3} cr
& Leftrightarrow frac{x}{y} = frac{{10}}{{13}} cr} $$

[#62] 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 ?
Correct Answer

(B) 216

Explanation

Solution: Let the number be x Then, $$eqalign{
& Leftrightarrow frac{2}{3}x - 50 = frac{1}{4}x + 40 cr
& Leftrightarrow frac{2}{3}x - frac{1}{4}x = 90 cr
& Leftrightarrow frac{{5x}}{{12}} = 90 cr
& Leftrightarrow x = left( {frac{{90 imes 12}}{5}}
ight) cr
& Leftrightarrow x = 216 cr} $$

[#63] The sum of two numbers is 25 and their difference is 13. Find their product :
Correct Answer

(B) 114

Explanation

Solution: Let the numbers be x and y Then, x + y = 25 and x - y = 13 $$eqalign{
& Leftrightarrow 4xy = {left( {x + y}
ight)^2} - {left( {x - y}
ight)^2} cr
& Leftrightarrow 4xy = {left( {25}
ight)^2} - {left( {13}
ight)^2} cr
& Leftrightarrow 4xy = 625 - 169 cr
& Leftrightarrow 4xy = 456 cr
& Leftrightarrow xy = 114 cr} $$

[#64] The sum of seven consecutive numbers is 175. What is the difference between twice the largest number and thrice the smallest number ?
Correct Answer

7

Explanation

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

[#65] The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. If the sum of the two digit of the number is 12, then what is the original number ?
Correct Answer

28

Explanation

Solution: Let ten's digit = x Then, unit's digit = (12 - x) $$ herefore left[ {10x + left( {12 - x}
ight)}
ight] - $$ xa0 xa0$$left[ {10left( {12 - x}
ight) + x}
ight]$$ xa0 $$ = 54$$ $$eqalign{
& Leftrightarrow 18x - 108 = 54 cr
& Leftrightarrow 18x = 162 cr
& Leftrightarrow x = 9 cr} $$ So, ten's digit = 9 and unit's digit = 3 Hence, original number = 93