Three consecutive integers are such that when they are taken in increasing…
2024
Three consecutive integers are such that when they are taken in increasing order and multiplied by 3, 5 and 2, respectively, they add up to 99. What is the sum of the original first and third integers?
Answer: C. 20 — Concept: Three consecutive integers can always be written as x, x+1, and x+2 for the smallest integer x. When each term of such a sequence is multiplied by…
- A.
16
- B.
18
- C.
20
- D.
24
Show answer & explanation
Correct answer: C
Concept: Three consecutive integers can always be written as x, x+1, and x+2 for the smallest integer x. When each term of such a sequence is multiplied by given constants and the results are summed to a stated total, the situation translates directly into a single linear equation in one variable, solved by expanding, collecting like terms, and isolating x.
Application:
Let the three consecutive integers, in increasing order, be x, x+1, and x+2.
Multiply each by 3, 5, and 2 respectively, as stated: 3x + 5(x+1) + 2(x+2).
Expand the products: 3x + 5x + 5 + 2x + 4.
Combine like terms: 10x + 9.
Set this equal to the given total and solve: 10x + 9 = 99, so 10x = 90, giving x = 9.
The three integers are 9, 10, and 11; the sum of the first and third is 9 + 11 = 20.
Cross-check: Substituting back into the original expression: 3(9) + 5(10) + 2(11) = 27 + 50 + 22 = 99, which matches the stated total, confirming the result.