Question 1

The sum of two numbers is 25 and their difference is 13. What is their product?

Accepted Answer

Problem restatement Given S = x + y = 25 and D = x − y = 13, find xy. Concept/Approach Use the identity S2 − D2 = 4xy. Step-by-Step calculation xy = (S2 − D2) ÷ 4 = (252 − 132) ÷ 4= (625 − 169) ÷ 4 = 456 ÷ 4 = 114 Verification/Alternative Explicit solution: x = (S + D)÷2 = 19, y = (S − D)÷2 = 6 ⇒ xy = 114. Common pitfalls Computing (S − D)2 instead of S2 − D2, or dividing by 2 instead of 4. Final Answer 114

Question 2

A two-digit number has its unit's digit 2 more than its ten's digit. If the product of the number and the sum of its digits is 144, what is the number?

Accepted Answer

Problem restatement Let the number be 10x + y, with y = x + 2. Also, (10x + y) × (x + y) = 144. Find the number. Given data y = x + 2 Product with digit-sum equals 144 Concept/Approach Express everything in x, solve the quadratic, and retrieve the digits. Step-by-Step calculation Number = 10x + (x + 2) = 11x + 2Digit-sum = x + (x + 2) = 2x + 2 = 2(x + 1)(11x + 2) × 2(x + 1) = 144 ⇒ (11x + 2)(x + 1) = 7211x2 + 13x + 2 = 72 ⇒ 11x2 + 13x − 70 = 0Discriminant = 132 + 4 × 11 × 70 = 3249 = 572x = (−13 + 57) ÷ 22 = 2 (valid digit); so y = 4Number = 24 Verification/Alternative Check: 24 × (2 + 4) = 24 × 6 = 144. Units digit (4) is 2 more than tens (2). Common pitfalls Forgetting that digits must be integers in 0–9; discarding the negative/invalid root is essential. Final Answer 24

Question 3

A 3-digit number has digit sum 10. The middle digit equals the sum of the other two, and reversing the digits increases the number by 99. What is the number?

Accepted Answer

Problem restatement Let the number be 100a + 10b + c with a + b + c = 10, b = a + c, and reverse − original = 99. Concept/Approach Translate the conditions into equations and solve for digits a, b, c. Step-by-Step calculation Reverse − original = (100c + 10b + a) − (100a + 10b + c) = 99(c − a) = 99Therefore, c − a = 1 ⇒ c = a + 1b = a + c = a + (a + 1) = 2a + 1Sum: a + b + c = a + (2a + 1) + (a + 1) = 4a + 2 = 10 ⇒ a = 2Then c = 3, b = 5 ⇒ Number = 253 Verification/Alternative Reversal 352; 352 − 253 = 99. Digit sum 2 + 5 + 3 = 10; middle digit 5 equals 2 + 3. Common pitfalls Setting reverse − original = 0 or 990 by misplacing place values; remember only hundreds and units exchange roles. Final Answer 253

Question 4

A number has two digits. If the digits are interchanged and the new number is added to the original, the result is divisible by what?

Accepted Answer

Problem restatement Let the original number be 10a + b. After swapping digits we get 10b + a. Show that their sum is always divisible by a fixed number. Concept/Approach Algebraic expression of the sum reveals a common factor. Step-by-Step calculation (10a + b) + (10b + a) = 11(a + b)Thus the sum is a multiple of 11 Common pitfalls Guessing 9 (digit-sum test) or 10 (last digit 0) without algebra; neither is generally true here. Final Answer 11

Question 5

A two-digit number has digit product 8. Adding 18 to the number reverses its digits. What is the number?

Accepted Answer

Problem restatement Let the number be 10a + b with ab = 8 and 10a + b + 18 = 10b + a. Find the number. Concept/Approach Use the reversal equation to relate a and b; combine with the product condition to solve integer digits. Step-by-Step calculation 10a + b + 18 = 10b + a ⇒ 9a − 9b = −18 ⇒ a − b = −2 ⇒ b = a + 2ab = 8 ⇒ a(a + 2) = 8Try digit possibilities: a = 2 ⇒ b = 4 works (2 × 4 = 8)Number = 24 Verification/Alternative 24 + 18 = 42, which is the reversal; product of digits = 8. Common pitfalls Taking a = 4, b = 2 (gives −2 = a − b = 2, contradiction). Final Answer 24

Question 6

The product of two numbers is 9375, and the quotient (larger ÷ smaller) is 15. What is the sum of the two numbers?

Accepted Answer

Problem restatement Let the smaller number be x and the larger be 15x. Given their product equals 9375; find x and hence their sum. Concept/Approach Use product and ratio to form an equation in one variable. Step-by-Step calculation (15x) × x = 9375 ⇒ 15x2 = 9375x2 = 9375 ÷ 15 = 625 ⇒ x = 25 (positive)Larger = 15x = 375Sum = 25 + 375 = 400 Verification/Alternative Product check: 25 × 375 = 9375. Quotient: 375 ÷ 25 = 15. Common pitfalls Taking square root as −25 (numbers are positive in this context) or mixing which is larger. Final Answer 400

Question 7

The difference between a two-digit number and the number obtained by interchanging its digits is 36. What is the difference between its two digits?

Accepted Answer

Problem restatement A two-digit number (tens digit a, units digit b) exceeds its reverse by 36. Find |a − b|. Given data Original number = 10a + b Reversed number = 10b + a Difference = (10a + b) − (10b + a) = 36 Concept/Approach Simplify the difference to get a linear equation in (a − b). Step-by-step calculation (10a + b) − (10b + a) = 9(a − b) = 36 a − b = 36 ÷ 9 = 4 Verification/Alternative If a − b = 4, then 9(a − b) = 36 exactly as given. Common pitfalls Using b − a instead of a − b; the absolute difference is 4 in any case. Final Answer 4

Question 8

The sum of the digits of a two-digit number is 15 and their difference is 3. What is the two-digit number?

Accepted Answer

Problem restatement Let the tens digit be a and the units digit be b. Given a + b = 15 and a − b = 3, find the number 10a + b. Given data a + b = 15 a − b = 3 Concept/Approach Solve the simultaneous linear equations. Step-by-step calculation Add the equations: (a + b) + (a − b) = 15 + 3 2a = 18 → a = 9 Then b = 15 − 9 = 6 Number = 10a + b = 10×9 + 6 = 96 Verification/Alternative Digits 9 and 6 have sum 15 and difference 3; number 96 fits. Common pitfalls Taking 69 instead of 96; the equations make tens digit 9, units 6. Final Answer 96

Question 9

If one-third of one-fourth of a number is 15, then three-tenths of that number is:

Accepted Answer

Problem restatement Given (1/3) of (1/4) of N equals 15. Find (3/10) of N. Given data (1/12)N = 15 Concept/Approach Compute N from the first relation, then take 3/10 of it. Step-by-step calculation N = 15 × 12 = 180 (3/10)N = (3/10) × 180 = 54 Verification/Alternative Check: (1/12) × 180 = 15, consistent. Final Answer 54

Question 10

Three times the first of three consecutive odd integers is 3 more than twice the third. What is the third integer?

Accepted Answer

Problem restatement Let the consecutive odd integers be x, x + 2, x + 4. Use the given linear relation to find x and then the third integer. Given data 3x = 2(x + 4) + 3 Concept/Approach Translate the verbal condition into an equation and solve. Step-by-step calculation 3x = 2x + 8 + 3 = 2x + 11 x = 11 Third integer = x + 4 = 15 Verification/Alternative Check: 3×11 = 33 and 2×15 + 3 = 33, so the condition holds. Final Answer 15

Question 11

The product of two numbers is 120 and the sum of their squares is 289. What is the sum of the numbers?

Accepted Answer

Problem restatement Let the numbers be a and b with ab = 120 and a2 + b2 = 289. Find a + b. Concept/Approach Use the identity (a + b)2 = a2 + b2 + 2ab. Step-by-step calculation (a + b)2 = 289 + 2×120 = 289 + 240 = 529 a + b = √529 = 23 Verification/Alternative 529 is 232. The sum could be −23 algebraically, but the context typically takes the positive sum. Final Answer 23

Question 12

A two-digit number exceeds the number formed by interchanging its digits by 36. If the ratio between its digits is 1 : 2, what is the difference between the sum and the difference of its digits?

Accepted Answer

Problem restatement Digits (tens = a, units = b) satisfy 9(a − b) = 36 ⇒ a − b = 4. The digits are in the ratio 1 : 2. Find (a + b) − (a − b). Given data a − b = 4 Digits in ratio 1 : 2 ⇒ the two digits are proportional to 1 and 2 (smaller : larger). Concept/Approach Since a − b > 0, the tens digit a is larger. Thus the pair must be (8, 4) up to the 1:2 ratio (4 : 8). Step-by-step calculation Take digits as 8 and 4 so that a − b = 4 and smaller : larger = 1 : 2 Sum = a + b = 12; Difference = a − b = 4 Required = (a + b) − (a − b) = 12 − 4 = 8 Verification/Alternative Check the interchange condition: 9(a − b) = 9×4 = 36 (satisfied). The ratio condition holds with digits {4, 8}. Common pitfalls Interpreting the ratio as a : b = 1 : 2 directly, which would contradict a − b = 4 (since that would make a < b). The ratio refers to values of the digits (smaller : larger). Final Answer 8

Question 13

The sum of the squares of three numbers is 138, and the sum of their pairwise products is 131. What is their sum?

Accepted Answer

Problem restatement Let the numbers be x, y, z. Given x2 + y2 + z2 = 138 and xy + yz + zx = 131. Find S = x + y + z. Concept/Approach Use the identity S2 = (x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx). Step-by-step calculation S2 = 138 + 2×131 = 138 + 262 = 400 S = √400 = 20 (taking the principal value) Verification/Alternative Algebraically S could be −20 as well, but unless specified otherwise, the conventional answer reported is 20. Common pitfalls Forgetting the factor of 2 on the sum of pairwise products. Final Answer 20

Question 14

Find the positive number which, when increased by 17, equals 60 times its reciprocal.

Accepted Answer

Problem restatement Find positive x such that x + 17 = 60 × (1/x). Concept/Approach Form a quadratic equation and solve using the discriminant. Step-by-step calculation x + 17 = 60/x x2 + 17x − 60 = 0 Discriminant D = 172 + 4×60 = 289 + 240 = 529 = 232 x = (−17 ± 23)/2 ⇒ x = 3 or x = −20 Positive solution ⇒ x = 3 Verification/Alternative 3 + 17 = 20 and 60 × (1/3) = 20; equality holds. Final Answer 3

Question 15

What is the sum of two consecutive even numbers if the difference of their squares is 84?

Accepted Answer

Problem restatement Let the even integers be x and x + 2. Given (x + 2)2 − x2 = 84, find x + (x + 2). Concept/Approach Use the identity (a + b)(a − b) = a2 − b2, or expand directly. Step-by-step calculation (x + 2)2 − x2 = 4x + 4 = 84 4(x + 1) = 84 → x + 1 = 21 → x = 20 Sum = x + (x + 2) = 20 + 22 = 42 Verification/Alternative Squares: 222 − 202 = 484 − 400 = 84, consistent. Final Answer 42

Question 16

Three numbers are in the ratio 1 : 2 : 3 and the sum of their cubes is 4500 .The smallest numbers is

Accepted Answer

5

Question 17

The largest 4 digit number exactly divisible by 88 is

Accepted Answer

To find the largest 4 digit number exactly divisible by 88, We should divide the largest possible 4 digit number by 88, and if we get any remainder than subtract it from that largest number. The largest possible 4 digit number is 9999 Now, 88) 9999 (113 88 _______ 119 88 _______ 319 264 _______ 55 _______ Therefore, the largest 4 digit number exactly divisible by 88 is given by 9999 - 55 = 9944.

Question 18

What is the place value of 6 in 64?

Accepted Answer

We know that, ● Each digit has a fixed position called its place. ● Each digit has a value depending on its place called the place value of the digit. ● The face value of a digit for any place in the given number is the value of the digit itself ● Place value of a digit = (face value of the digit) × (value of the place). Hence, the place value of 6 in 64 = 6 x 10 = 60.

Question 19

In a two-digit number, the digit in the unit's place is more than twice the digit in ten's place by 1. If the digits in the unit's place and the ten's place are interchanged,difference between the newly formed number and the oiginal number is less t…

Accepted Answer

Let the ten's digit be x. Then, unit's digit = 2x + 1. [10x + (2x + 1)] - [{10 (2x + 1) + x} - {10x + (2x + 1)}] = 1 <=> (12x + 1) - (9x + 9) = 1 <=> 3x = 9, x = 3. So, ten's digit = 3 and unit's digit = 7. Hence, original number = 37.

Question 20

Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is :

Accepted Answer

Let the three integers be x, x + 2 and x + 4. Then, 3x = 2 (x + 4) + 3 <=> x = 11. Third integer = x + 4 = 15.

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