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📝 Finding unknown numbers in algebra (10 MCQs)

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

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What is Finding unknown numbers in algebra?

Definition:
Finding unknown numbers in algebra involves solving equations where the unknown is represented by a variable. The problem gives a relationship (e.g., sum, difference, product) involving the unknown, and you use algebraic techniques to isolate the variable and determine its value.

Working:
For example, 'A number plus 7 is 20.' Let the number be $x$ . Equation: $x + 7 = 20$ . Subtract 7: $x = 13$ . The unknown number is 13.

Example:
Problem: 'When a number is multiplied by 4, the result is 32.' Let $x$ be the number. Equation: $4x = 32$ . Divide by 4: $x = 8$ .

Reason:
This skill is fundamental because it allows students to solve for missing values in various mathematical and real-world contexts.

Easy

Medium

Hard

📝 All Finding unknown numbers in algebra MCQs

Q1. A number is increased by 7 and the result is three times the original number minus 9. Which equation correctly models the situation if $x$ represents the unknown number?

x+7=3x-9

✅

x+7=3(x-9)

7x=3x-9

x-7=3x+9

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The phrase increased by 7 means $x+7$ , while three times the original number minus 9 means $3x-9$ . Translating both expressions carefully gives $x+7=3x-9$ , preserving the intended order of operations.

Q2. A two-digit number has digits whose sum is 11. The tens digit is 3 more than the units digit. Which number satisfies these conditions?

A.47

B.56

C.74 ✅

D.83

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Let the units digit be $x$ . The tens digit is $x+3$ , and their sum gives $x+(x+3)=11$ . Thus $2x=8$ , so $x=4$ , and the tens digit is 7, producing 74.

Q3. A teacher says, "Five less than twice a number is 19." A student writes $2(x-5)=19$ . What is the main modeling error?

A.The student should subtract 19 first.

B.The phrase "twice a number" should be written

2x

before subtracting 5. ✅

C.The number should be divided by 5 first.

D.The equation should contain

x+5

instead of

x-5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The wording means first take twice the number, $2x$ , and then subtract 5, giving $2x-5=19$ . Writing $2(x-5)$ incorrectly makes the subtraction occur before doubling.

Q4. A number is 8 less than another number. Their sum is 64. Which strategy most efficiently finds the smaller number?

A.Let the smaller be

x

, write

x+(x+8)=64

, then solve. ✅

B.Let the smaller be

x

, write

x+(x-8)=64

, then solve.

C.Divide 64 by 8 because the numbers differ by 8.

D.Subtract 8 from 64 and use the result as the smaller number.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Representing the smaller number as $x$ makes the larger $x+8$ . Their sum becomes $x+(x+8)=64$ , which leads to $2x+8=64$ and $x=28$ . This method directly models both conditions.

Q5. A number line shows two unknown numbers equally spaced around 25. Their distance from 25 is 9 units on each side. What is the sum of the two unknown numbers?

A.16

B.25

C.41

D.50 ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The two numbers are $25-9=16$ and $25+9=34$ . Their sum is $16+34=50$ . The equal spacing means the midpoint is 25, so the sum is twice the midpoint, $2(25)=50$ .

Q6. A graph represents $y=2x+6$ and another line represents $y=54-x$ . Their intersection represents two numbers whose sum is 54, with one number twice the other plus 6. What are the numbers?

A.14 and 40

B.16 and 38 ✅

C.18 and 36

D.20 and 34

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: At the intersection, $2x+6=54-x$ . Thus $3x=48$ , giving $x=16$ . The corresponding value is $54-16=38$ . The graph's intersection therefore identifies the pair 16 and 38.

Q7. Two numbers differ by 12. The larger number is four times the smaller number. Which pair is possible, and why?

A.4 and 16, because

16-4=12

✅

B.6 and 24, because

24-6=18

C.8 and 20, because

20-8=12

D.12 and 48, because

48-12=36

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Let the smaller number be $x$ . The larger is $4x$ , and their difference is $12$ , so $4x-x=12$ . Hence $3x=12$ , giving $x=4$ and the larger number 16.

Q8. A student solves the equation $3x+8=2x+21$ and obtains $x=13$ . Which verification best confirms the result?

A.Both sides become 47. ✅

B.Both sides become 39.

C.Only the left side becomes 47.

D.Only the right side becomes 39.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substituting $x=13$ into the original equation gives $3(13)+8=47$ and $2(13)+21=47$ . Since both sides have the same value, $x=13$ satisfies the original relationship.

Q9. A puzzle asks for two positive integers whose product is 48 and whose sum is 14. Which pair should be selected?

A.4 and 12

B.5 and 9

C.6 and 8 ✅

D.7 and 7

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The factor pairs of 48 include $1,48$ , $2,24$ , $3,16$ , $4,12$ , and $6,8$ . Only $6+8=14$ , while $6\times8=48$ , so the pair satisfying both conditions is 6 and 8.

Q10. A student models a number problem with $x+(x+5)=41$ and obtains $x=18$ . Another student says the larger number should instead be $x+10$ . Which conclusion is correct?

A.The first model is correct if the numbers differ by 5. ✅

B.The second model is correct because the total is 41.

C.Both models are equivalent for every value of

x

D.Neither model can represent two unknown numbers.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If one number is 5 greater than the other, the larger must be $x+5$ , not $x+10$ . The equation gives $2x+5=41$ , so $x=18$ and the numbers are 18 and 23. Verification confirms the first model.

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