Integer Math Problems: Practice Questions with Step-by-Step Solutions
Integer word problems require translating verbal statements into algebraic equations. These practice problems cover consecutive integers, consecutive even and odd integers, and product conditions. Try each problem first, then open the solution to check your work.
Review: Representing Consecutive Integers
- Consecutive integers: \(x,\; x+1,\; x+2,\; \dots\)
- Consecutive even integers: \(x,\; x+2,\; x+4,\; \dots\) (with \(x\) even)
- Consecutive odd integers: \(x,\; x+2,\; x+4,\; \dots\) (with \(x\) odd)
Translate the words into an equation, solve, then verify by substitution.
Practice Problems and Solutions
Problem 1: Two consecutive integers
Question: Find two consecutive integers whose sum is equal to 129.
Solution:
Let \(x\) and \(x + 1\) be the two numbers. Their sum is 129, so:
\[ x + (x + 1) = 129 \]Combine like terms:
\[ 2x + 1 = 129 \] \[ 2x = 128 \implies x = 64 \]The two numbers are \(64\) and \(65\).
Check: \(64 + 65 = 129\). ✔
Problem 2: Three consecutive integers
Question: Find three consecutive integers whose sum is equal to 366.
Solution:
Let \(x\), \(x + 1\), and \(x + 2\) be the three numbers. Their sum is 366, so:
\[ x + (x + 1) + (x + 2) = 366 \] \[ 3x + 3 = 366 \] \[ 3x = 363 \implies x = 121 \]The three numbers are \(121\), \(122\), and \(123\).
Problem 3: Three consecutive even integers
Question: The sum of three consecutive even integers is 84. Find the numbers.
Solution:
Consecutive even integers differ by 2. Let \(x\), \(x + 2\), and \(x + 4\) be the numbers. Their sum is 84, so:
\[ x + (x + 2) + (x + 4) = 84 \] \[ 3x + 6 = 84 \] \[ 3x = 78 \implies x = 26 \]The three even numbers are \(26\), \(28\), and \(30\).
Check: \(26 + 28 + 30 = 84\). ✔
Problem 4: Odd integers and multipliers
Question: The sum of an odd integer and twice its consecutive is 3757. Find the number.
Solution:
Consecutive odd integers differ by 2. Let \(x\) be the odd integer and \(x + 2\) its consecutive. The condition gives:
\[ x + 2(x + 2) = 3757 \] \[ x + 2x + 4 = 3757 \] \[ 3x + 4 = 3757 \] \[ 3x = 3753 \implies x = 1251 \]Check: \(1251 + 2(1253) = 1251 + 2506 = 3757\). ✔
Problem 5: Complex relationships among odd integers
Question: The sum of the first and third of three consecutive odd integers is 131 less than three times the second integer. Find the three integers.
Solution:
Let \(x\), \(x + 2\), and \(x + 4\) be the three consecutive odd integers.
- Sum of first and third: \(x + (x + 4)\)
- 131 less than three times the middle: \(3(x + 2) - 131\)
Set them equal:
\[ x + (x + 4) = 3(x + 2) - 131 \] \[ 2x + 4 = 3x + 6 - 131 \] \[ 2x + 4 = 3x - 125 \] \[ x = 129 \]The three numbers are \(129\), \(131\), and \(133\).
Check: \(129 + 133 = 262\); \(3(131) - 131 = 262\). ✔
Problem 6: Product of consecutive odd integers
Question: The product of two consecutive odd integers is 675. Find the two integers.
Solution:
Let \(x\) and \(x + 2\) be the two odd integers. Their product is 675:
\[ x(x + 2) = 675 \] \[ x^2 + 2x - 675 = 0 \]Factor (two numbers that multiply to \(-675\) and add to \(2\)):
\[ (x - 25)(x + 27) = 0 \] \[ x = 25 \quad \text{or} \quad x = -27 \]The two possible pairs are:
\[ 25 \text{ and } 27 \quad \textbf{or} \quad -27 \text{ and } -25 \]Problem 7: Summing pairs of consecutive even integers
Question: Find four consecutive even integers so that the sum of the first two added to twice the sum of the last two equals 742.
Solution:
Let the four consecutive even integers be \(x\), \(x+2\), \(x+4\), \(x+6\).
- Sum of first two: \(x + (x+2)\)
- Twice sum of last two: \(2\big((x+4)+(x+6)\big) = 4x + 20\)
Set the total to 742:
\[ x + (x+2) + 4x + 20 = 742 \] \[ 6x + 22 = 742 \] \[ 6x = 720 \implies x = 120 \]The four consecutive even integers are \(120\), \(122\), \(124\), and \(126\).
Check: \((120+122) + 2(124+126) = 242 + 500 = 742\). ✔
Problem 8: Complex relationships among three odd integers
Question: When the smallest of three consecutive odd integers is added to four times the largest, it produces a result 729 more than four times the middle integer. Find the numbers and check your answer.
Solution:
Let \(x\), \(x + 2\), and \(x + 4\) be the three consecutive odd integers.
- Smallest plus four times largest: \(x + 4(x+4)\)
- 729 more than four times middle: \(729 + 4(x+2)\)
Set them equal:
\[ x + 4(x+4) = 729 + 4(x+2) \] \[ x + 4x + 16 = 729 + 4x + 8 \] \[ 5x + 16 = 4x + 737 \] \[ x = 721 \]The three odd integers are \(721\), \(723\), and \(725\).
Check:
- \(721 + 4(725) = 721 + 2900 = 3621\)
- \(729 + 4(723) = 729 + 2892 = 3621\)
Both sides equal \(3621\). ✔
Key Takeaways
- Translate the verbal statement into one clear algebraic equation before solving.
- Consecutive integers differ by 1; consecutive even or odd integers differ by 2.
- Quadratic word problems may have two valid answer pairs (e.g., positive and negative).
- Always substitute your answer back into the original condition to verify.