Question 1

When one-seventh of a number is subtracted from the number itself, the result equals the sum of the angles in a triangle (180 degrees). What is the number?

Accepted Answer

Introduction / Context: This is a linear equation in one variable with a familiar constant (the angle sum of a triangle). Converting the verbal statement into an algebraic equation and solving it yields the unique value of the number. Given Data / Assumptions: Let the number be N. The sum of angles in a triangle is 180 degrees. Condition: N − (1/7)N = 180. Concept / Approach: Combine like terms on the left: N − (1/7)N = (6/7)N. Then isolate N by dividing 180 by the fractional coefficient. This is a direct application of fraction arithmetic. Step-by-Step Solution: Write the equation: N − N/7 = 180.Compute the coefficient: (6/7)N = 180.Isolate N: N = 180 * (7/6) = 30 * 7 = 210. Verification / Alternative check: Check: 210 − (1/7)*210 = 210 − 30 = 180, which satisfies the given relation. Why Other Options Are Wrong: 224, 140, 350, and 196 do not produce 180 after subtracting one-seventh of themselves. Common Pitfalls: Subtracting 1/7 incorrectly (e.g., taking N − 1/7 instead of N − N/7), or miscomputing 180 * 7/6. Final Answer: 210

Question 2

In a division operation, the divisor equals ten times the quotient and also equals five times the remainder. If the remainder is 46, find the dividend.

Accepted Answer

Introduction / Context: This problem connects the four elements of a division algorithm: dividend, divisor, quotient, and remainder. Using the given proportional relationships, you can determine the divisor and quotient, then reconstruct the dividend through the standard identity Dividend = Divisor * Quotient + Remainder. Given Data / Assumptions: Remainder R = 46. Divisor D = 5 * R. Divisor D = 10 * Quotient Q. Division identity: Dividend N = D * Q + R. Concept / Approach: First compute D from the remainder. Then use D = 10Q to find Q. Finally, apply N = DQ + R. This sequence avoids guesswork and keeps arithmetic simple and exact. Step-by-Step Solution: Compute D: D = 5 * 46 = 230.Find Q from D = 10Q → Q = D / 10 = 23.Compute N: N = D * Q + R = 230 * 23 + 46.Evaluate: 230 * 23 = 230 * (20 + 3) = 4600 + 690 = 5290; then N = 5290 + 46 = 5336. Verification / Alternative check: Check proportions: D = 230 is five times 46 and ten times 23. Substituting into N = DQ + R reproduces the given remainder exactly. Why Other Options Are Wrong: 5388, 5343, 5391, and 5328 do not satisfy the division identity with D = 230, Q = 23, and R = 46. Common Pitfalls: Confusing which term is 10 times which (mistaking Q = 10D), or omitting the remainder when reconstructing the dividend. Final Answer: 5336

Question 3

Consecutive odd integers — product given The product of two consecutive odd numbers is 6723. Determine the greater of the two odd integers.

Accepted Answer

Introduction / Context: This question checks understanding of consecutive odd numbers and converting a word statement into a simple quadratic equation. Consecutive odd integers differ by 2, and their product is supplied. The task is to retrieve the larger integer directly and verify it. Given Data / Assumptions: The two odd integers are consecutive: let them be n and n + 2. Their product equals 6723: n(n + 2) = 6723. We seek the greater integer (n + 2). Concept / Approach: Form a quadratic equation from the product relation and solve it using the discriminant method. Because the numbers are integers and consecutive odds, the quadratic should factor nicely or yield an integer root after applying the square root to the discriminant. Step-by-Step Solution: Start: n(n + 2) = 6723 → n^2 + 2n − 6723 = 0.Compute discriminant: D = 2^2 + 4*6723 = 4 + 26892 = 26896.Recognize 26896 = 164^2, so roots are integers.n = (−2 + 164)/2 = 81 (take positive root).Consecutive odd numbers: 81 and 83 → greater = 83. Verification / Alternative check: Multiply 81 * 83 = 81*(80 + 3) = 6480 + 243 = 6723, which matches the given product. Therefore, 83 is correct. Why Other Options Are Wrong: 89, 85, and 91 are not paired with a neighboring odd to give 6723; 81 is the smaller integer, not the greater one requested. Common Pitfalls: Using (n, n + 1) instead of odd spacing of 2; arithmetic errors in discriminant; forgetting to extract the greater value after solving for n. Final Answer: 83

Question 4

Sum of four consecutive even integers The sum of four consecutive even numbers is 284. What is the smallest of these even integers?

Accepted Answer

Introduction / Context: Consecutive even integers increase by 2 each time. Summation of such a group forms a quick linear equation. This tests modeling skills and careful arithmetic. Given Data / Assumptions: The four even numbers are consecutive: a, a + 2, a + 4, a + 6. Their total sum is 284. We must find the smallest, a. Concept / Approach: Write the sum in terms of a and equate to 284. Because the common difference is 2, the sum is 4a + 12. Solve the resulting linear equation for a. Step-by-Step Solution: Sum: a + (a + 2) + (a + 4) + (a + 6) = 284.Combine like terms: 4a + 12 = 284.Subtract 12: 4a = 272.Divide by 4: a = 68. Verification / Alternative check: Numbers are 68, 70, 72, 74. Sum = 68 + 70 + 72 + 74 = (68 + 74) + (70 + 72) = 142 + 142 = 284. Checks out. Why Other Options Are Wrong: 66 and 70 are near misses that give totals not equal to 284; 72 and 74 are not the smallest in the valid set. Common Pitfalls: Using odd increments of 1 instead of 2; arithmetic slips when subtracting 12; misidentifying the smallest number after solving. Final Answer: 68

Question 5

Two-digit number with digit sum and digit difference A two-digit number has digits that sum to 14 and differ by 2. What is the product of the two digits of the number?

Accepted Answer

Introduction / Context: Digit problems often give the sum and difference of two digits. This makes it straightforward to solve for the individual digits using simultaneous linear equations and then compute the desired product. Given Data / Assumptions: Let the digits be L (larger) and S (smaller). L + S = 14. L − S = 2. We want L * S. Concept / Approach: Add and subtract the equations to isolate L and S. This is a standard method for two equations in two unknowns. After finding the digits, multiply them for the requested product. Step-by-Step Solution: Add equations: (L + S) + (L − S) = 14 + 2 → 2L = 16 → L = 8.Find S: S = 14 − L = 14 − 8 = 6.Product: L * S = 8 * 6 = 48. Verification / Alternative check: Digits 8 and 6 satisfy both conditions: sum 14, difference 2. The product 48 follows directly. Why Other Options Are Wrong: 56 and 45 arise from mixing digits or arithmetic slips; 40 is another near miss; “Couldn't be determined” is incorrect because the system has a unique solution. Common Pitfalls: Interchanging sum and difference equations or assuming the digits can be in any order without checking the difference sign. Final Answer: 48

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