Problems On Numbers - Study Mode

[#91] The difference between two integers is 5. Their product is 500. Find the numbers.
Correct Answer

(B) 20, 25

Explanation

Solution: Let the integers be x and (x + 5) Then, $$eqalign{
& Leftrightarrow xleft( {x + 5}
ight) = 500 cr
& Leftrightarrow {x^2} + 5x - 500 = 0 cr
& Leftrightarrow left( {x + 25}
ight)left( {x - 20}
ight) = 0 cr
& Leftrightarrow x = 20 cr} $$ So, the numbers are 20 and 25

[#92] What is the sum of two consecutive even numbers, the difference of whose squares is 84 ?
Correct Answer

(C) 42

Explanation

Solution: Let the numbers be x and x + 2 Then, (x + 2) 2 - x 2 = 84 ⇔ 4x + 4 = 84 ⇔ 4x = 80 ⇔ x = 20 ∴ Required sum = x + (x + 2) = 2x + 2 = 42

[#93] A certain number of two digits is three times the sum of its digits and if 45 be added to it, the digits are reversed. The number is ?
Correct Answer

(B) 27

Explanation

Solution: Let the ten's digit be x and unit's digit be y Then, $$eqalign{
& Rightarrow 10x + y = 3left( {x + y}
ight) cr
& Rightarrow 7x - 2y = 0.....(i) cr
& 10x + y + 45 = 10y + x cr
& Rightarrow y - x = 5.....(ii) cr} $$ Solving (i) and (ii), we get : x = 2 and y = 7 ∴ Required number = 27

[#94] If the product of two numbers is 5 and one of the number is $$frac{3}{2}$$, then sum of two numbers is :
Correct Answer

(C) $$4frac{5}{6}$$

Explanation

Solution: Let two numbers be a and b Given ab = 5 and a = $$frac{3}{2}$$ $$eqalign{
& Rightarrow b = frac{5}{a} cr
& Rightarrow b = frac{5}{{frac{3}{2}}} cr
& Rightarrow b = frac{{5 imes 2}}{3} cr
& Rightarrow b = frac{{10}}{3} cr} $$ Required sum of : ⇒ a + b = $$frac{{3}}{2}$$ + $$frac{{10}}{3}$$ L.C.M. of 2 and 3 = 6 ⇒ a + b = $$frac{{9 + 20}}{6}$$ ⇒ a + b = $$frac{{29}}{6}$$ ⇒ a + b = $$4frac{{5}}{6}$$

[#95] If a number is added to two-fifths of itself, the value so obtained is 455. What is the number ?
Correct Answer

(A) 325

Explanation

Solution: Let the number be x Then, $$eqalign{
& Leftrightarrow x + frac{2}{5}x = 455 cr
& Leftrightarrow frac{7}{5}x = 455 cr
& Leftrightarrow x = left( {frac{{455 imes 5}}{7}}
ight) cr
& Leftrightarrow x = 325 cr} $$