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📝 Consecutive odd integers (10 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 10 questions available

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What is Consecutive odd integers?

Definition:
Consecutive odd integers are odd numbers that follow sequentially, differing by 2 (e.g., 1, 3, 5, or -3, -1, 1). If the first odd integer is $x$ , the next are $x+2$ , $x+4$ , etc. These are solved similarly to consecutive even integers.

Working:
Example: 'The sum of three consecutive odd integers is 75.' Let the first be $x$ , so numbers are $x, x+2, x+4$ . Equation: $x + (x+2) + (x+4) = 75$ , $3x+6=75$ , $3x=69$ , $x=23$ . The integers are 23, 25, 27.

Example:
Problem: 'Find two consecutive odd integers whose sum is 48.' Let $x$ and $x+2$ . Equation: $x + (x+2) = 48$ , $2x+2=48$ , $2x=46$ , $x=23$ . Integers: 23 and 25.

Reason:
This develops pattern recognition for odd numbers and reinforces the concept that consecutive odds also have a common difference of 2.

Easy

Medium

Hard

📝 All Consecutive odd integers MCQs

Q1. Three consecutive odd integers have a sum of 75. Which set of integers satisfies the condition?

A.21, 25, 29

B.23, 25, 27 ✅

C.19, 25, 31

D.21, 23, 31

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Consecutive odd integers differ by exactly 2. Testing the middle value 25 gives the numbers 23, 25, and 27, whose sum is 75. The other choices either have incorrect spacing or do not produce the required total.

Q2. The sum of three consecutive odd integers is 147. A student starts with $x$ , $x+2$ , and $x+4$ . What should the student do next to find the integers?

A.Solve

3x+6=147

, giving

x=47

✅

B.Solve

3x+4=147

, giving

x=47

C.Solve

3x+6=147

, giving

x=49

D.Solve

3x=147

, giving

x=49

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Adding $x+(x+2)+(x+4)$ produces $3x+6$ . Setting this equal to 147 gives $3x=141$ , so $x=47$ . The three integers are therefore 47, 49, and 51.

Q3. A rectangular garden has a length and width that are consecutive odd integers. If its perimeter is 64 meters, what are the dimensions?

A.15 m by 17 m ✅

B.13 m by 19 m

C.17 m by 19 m

D.15 m by 19 m

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Let the dimensions be $x$ and $x+2$ . The perimeter equation is $2x+2(x+2)=64$ . This gives $4x+4=64$ , so $x=15$ , making the dimensions 15 m and 17 m.

Q4. A student claims that $x-2$ , $x$ , and $x+2$ cannot represent consecutive odd integers because one expression might become even. What is the best evaluation?

A.The claim is correct because subtraction changes parity.

B.The claim is correct unless

x

is positive.

C.The claim is incorrect because if

x

is odd, all three expressions remain odd and differ by 2. ✅

D.The claim is incorrect because consecutive integers always differ by 1.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Adding or subtracting 2 preserves parity. Therefore, when $x$ is odd, $x-2$ , $x$ , and $x+2$ are all odd and are consecutive within the odd-integer sequence. The student's reasoning confuses consecutive integers with consecutive odd integers.

Q5. A student models three consecutive odd integers as $x$ , $x+1$ , and $x+2$ , then solves a sum problem successfully but obtains one even integer. What is the fundamental modeling error?

A.The student should have multiplied all terms.

B.The student used a difference of 1 instead of the required difference of 2. ✅

C.The student should have started with an even value.

D.The student should have used

x-1

x

, and

x+1

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Consecutive odd integers increase by 2 rather than 1. Using $x$ , $x+1$ , and $x+2$ models consecutive integers and inevitably includes an even number when $x$ is odd. The incorrect step is the assumed difference of 1.

Q6. A number line marks three equally spaced points at 17, 19, and 21. A second student says these are consecutive integers because no integers lie between adjacent marked points. Which response is most accurate?

A.The student is correct because equal spacing means consecutive integers.

B.The student is correct because 17, 19, and 21 have no fractions between them.

C.The student is incorrect because consecutive odd integers must differ by 3.

D.The student is incorrect because consecutive integers differ by 1, while these marked values differ by 2. ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The points 17, 19, and 21 are consecutive odd integers because each neighboring pair differs by 2. They are not consecutive integers, since consecutive integers differ by exactly 1. The spacing on the number line reveals the distinction.

Q7. A graph plots the possible middle value $x$ against the sum $S$ of three consecutive odd integers modeled as $x-2$ , $x$ , and $x+2$ . Which relationship should the graph show?

A.A horizontal line

S=3

B.A line with equation

S=3x

✅

C.A parabola

S=x^2+4

D.A line with equation

S=x+4

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Adding $x-2$ , $x$ , and $x+2$ cancels the constant terms, leaving $S=3x$ . Therefore the graph is a straight line through the origin with slope 3, reflecting the direct relationship between the middle integer and the total.

Q8. A theater numbers three adjacent reserved seats using consecutive odd numbers. Their numbers add to 219. If the organizer instead starts with the largest number and works backward, what set should be obtained?

A.71, 73, 75 ✅

B.69, 71, 79

C.71, 73, 77

D.69, 73, 77

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Let the middle seat be $x$ . Then the three consecutive odd numbers are $x-2$ , $x$ , and $x+2$ . Since their sum is 219, $3x=219$ , giving $x=73$ . Thus the seats are 71, 73, and 75.

Q9. Two methods solve a problem involving three consecutive odd integers with a sum of 105. Method 1 uses $x,x+2,x+4$ . Method 2 uses $x-2,x,x+2$ . Which comparison is correct?

A.Only Method 1 works because the first integer must be

x

B.Only Method 2 works because

x

must be the middle integer.

C.Both methods work, but they assign different meanings to

x

and produce the same integer set. ✅

D.Neither method works because three odd integers cannot have an odd sum.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both models correctly represent three consecutive odd integers. Method 1 treats $x$ as the smallest, while Method 2 treats $x$ as the middle. The resulting values are the same set, even though the variable represents different integers.

Q10. The sum of five consecutive odd integers is 245. Without solving for every integer separately, which value must be the middle integer, and why?

A.47, because five terms are involved

B.49, because the sum divided by the number of terms gives the middle value ✅

C.51, because the largest value is closest to the average

D.53, because the common difference is 2

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For an odd number of equally spaced terms, the average equals the middle term. Since $245\div5=49$ , the middle integer is 49. The complete sequence would be 45, 47, 49, 51, and 53.

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