PROBLEMS ON NUMBERS(LEVEL - 1)
Q1–10 of 50If a number of two digits is k times the sum of its digits, then the number formed by interchanging the digits is the sum of the digits multiplied by
Step 1: Let the original two-digit number be 10x+y. Given 10x+y=k(x+y).
Step 2: The interchanged number is 10y+x. We need to find m such that 10y+x=m(x+y).
Step 3: Add both equations: (10x+y)+(10y+x)=(k+m)(x+y)⟹11(x+y)=(k+m)(x+y).
Step 4: Cancel (x+y): 11=k+m⟹m=11−k.
Answer: (b) 11−k
A two-digit number exceeds the sum of the digits of that number by 18. If the digit at the unit's place is double the digit in the ten's place, what is the number?
tep 1: Let the number be $10x + y$. Given: $(10x + y) - (x + y) = 18 \implies 9x = 18 \implies x = 2$.
Step 2: Given $y = 2x \implies y = 2(2) = 4$.
Step 3: The number is $10(2) + 4 = 24$.
Answer: (a) 24
The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?
Step 1: Let the digits be $x$ and $y$. Given $x + y = 15$ and $\vert{}x - y\vert{} = 3$.
Step 2: Case 1: $x - y = 3 \implies 2x = 18 \implies x = 9, y = 6$. The number is 96.
Step 3: Case 2: $y - x = 3 \implies 2y = 18 \implies y = 9, x = 6$. The number is 69.
Step 4: Since we cannot uniquely determine whether the number is 69 or 96, it cannot be determined.
Answer: (d) Cannot be determined
A two-digit number is 7 times the sum of its digits. The number that is formed by reversing its digits is 18 less than the original number. What is the number?
Step 1: Let the number be $10x + y$. Given $10x + y = 7(x + y) \implies 3x = 6y \implies x = 2y$.
Step 2: Original number minus reversed number is 18:
Step 3: Substitute $x = 2y$: $2y - y = 2 \implies y = 2$, so $x = 4$.
Step 4: The original number is 42.
Answer: (a) 42
If the digit in the unit's place of a two-digit number is halved and the digit in the ten's place is doubled, the number thus obtained is equal to the number obtained by interchanging the digits. Which of the following is definitely true?
Step 1: Let the original number be $10x + y$.
Step 2: According to the problem: $10(2x) + \frac{y}{2} = 10y + x$.
Step 3: Multiply by 2: $40x + y = 20y + 2x \implies 38x = 19y \implies y = 2x$.
Step 4: The digit in the unit's place ($y$) is twice the digit in the ten's place ($x$).
Answer: (d) Digit in the unit's place is twice the digit in the ten's place.
In a two-digit number, if it is known that its unit's digit exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is
Step 1: Let ten's digit be $x$, then unit's digit is $x + 2$.
Step 2: The number is $10x + (x + 2) = 11x + 2$.
Step 3: Sum of digits $= x + (x + 2) = 2x + 2$.
Step 4: Given: $(11x + 2)(2x + 2) = 144 \implies (11x + 2)(x + 1) = 72$.
Step 5: Test $x = 2$: $(11(2) + 2)(2 + 1) = 24 \times 3 = 72$.
Step 6: So ten's digit is 2, unit's digit is 4, making the number 24.
Answer: (a) 24
A number consists of two digits. If the digits interchange places and the new number is added to the original number, then the resulting number will be divisible by
Step 1: Original number $= 10x + y$, Reversed number $= 10y + x$.
Step 2: Sum $= (10x + y) + (10y + x) = 11x + 11y = 11(x + y)$.
Step 3: The result is always a multiple of 11.
Answer: (d) 11
The sum of the digits of a two-digit number is 9 less than the number. Which of the following digits is at unit's place of the number?
Step 1: Let the number be $10x + y$.
Step 2: Given: $(10x + y) - (x + y) = 9 \implies 9x = 9 \implies x = 1$.
Step 3: Ten's digit $x = 1$, but $y$ (unit's digit) can be any digit from 0 to 9. Therefore, unit's digit is indeterminate.
Answer: (d) Data inadequate
The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
Step 1: Difference between original and reversed number is $9(x - y) = 36$.
Step 2: Divide by 9: $x - y = 4$.
Answer: (b) 4
The difference between a two-digit number and the number obtained by interchanging the two digits is 63. Which is the smaller of the two numbers?
Step 1: Difference between numbers is $9(x - y) = 63 \implies x - y = 7$.
Step 2: Possible pairs for $(x, y)$ are $(9, 2)$ or $(8, 1)$.
Step 3: The numbers could be 92/29 or 81/18. The exact smaller number cannot be determined.
Answer: (d) Cannot be determined