← Back to 3. Mathematical Models in Algebra

📝 How to Solve number problems (12 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 12 questions available

←Previous Topic:How to Use a problem solving strategy for word problems📝 How to Solve number problemsTranslate English sentences into algebraic equations:Next Topic→

What is How to Solve number problems?

Definition:
Solving number problems involves finding unknown numerical values based on given relationships described in words. These problems typically involve operations like addition, subtraction, multiplication, or division and are solved by translating the verbal description into a linear equation with one variable.

Working:
For example, 'If you add 5 to a number and get 17, find the number.' Let the number be $x$ . Equation: $x + 5 = 17$ . Solve: $x = 12$ . The number is 12.

Example:
Problem: 'Twice a number is 36.' Let $x$ be the number. Equation: $2x = 36$ . Divide by 2: $x = 18$ . The number is 18.

Reason:
Number problems are the simplest application of algebra, building foundational skills for more complex word problems and real-life calculations.

Easy

Medium

Hard

📝 All How to Solve number problems MCQs

Q1. A number is increased by 7 and then doubled. The result is 34. Which equation correctly models the situation and gives the original number?

2x+7=34

2(x+7)=34

✅

x+14=34

2x-7=34

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The phrase increased by 7 means 7 is added to the original number before the doubling occurs. Therefore the model is $2(x+7)=34$ , giving $x+7=17$ and $x=10$ .

Q2. Two consecutive integers have a sum of 57. If the smaller integer is represented by $x$ , which expression represents the larger integer?

x-1

x+1

✅

2x+1

57-x+1

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Consecutive integers differ by exactly 1. If $x$ represents the smaller integer, the next integer must be $x+1$ . This representation is essential before forming the equation $x+(x+1)=57$ .

Q3. The sum of three consecutive integers is 72. Which reasoning correctly determines the middle integer without unnecessarily testing many values?

A.Let the integers be

x,x+1,x+2

, then solve

3x+3=72

✅

B.Let the integers be

x,x+2,x+4

, then solve

3x+6=72

C.Let all three integers equal

x

, then solve

3x=72

D.Let the integers be

x-1,x,x+1

, then solve

2x+1=72

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Representing consecutive integers as $x,x+1,x+2$ preserves their required spacing. Solving $x+(x+1)+(x+2)=72$ gives $3x+3=72$ , so $x=23$ and the middle integer is $24$ .

Q4. The sum of two numbers is 41, and the larger number is 5 more than the smaller. Which pair satisfies both conditions?

A.17 and 24

B.18 and 23 ✅

C.19 and 22

D.16 and 25

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Let the smaller number be $x$ . The larger is $x+5$ , so $x+(x+5)=41$ . This gives $2x=36$ , hence $x=18$ and the larger number is $23$ , satisfying both conditions.

Q5. A student models a problem about two numbers as $x+(x+8)=50$ but obtains $x=21$ . What should the student conclude?

A.The answer is correct because

21+29=50

B.The equation should have been

x+(x-8)=50

C.The arithmetic is incorrect because solving the equation gives

x=21

, but the paired number should be

29

✅

D.The problem has no solution because 50 is even

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The equation $x+(x+8)=50$ gives $2x+8=50$ , so $2x=42$ and $x=21$ . The second number is $29$ , and $21+29=50$ . The result is therefore valid.

Q6. A two-digit number has digits whose sum is 11. Reversing the digits increases the number by 27. Which number satisfies these conditions?

A.38

B.47 ✅

C.56

D.65

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Let the tens digit be $a$ and the units digit be $b$ . Then $a+b=11$ . Reversal changes the value by $9(b-a)=27$ , so $b-a=3$ . Solving gives $a=4,b=7$ , producing $47$ .

Q7. A positive integer is multiplied by 4, and then 6 is subtracted. The result equals three times the original integer plus 9. What is the integer?

A.3

B.12

C.15 ✅

D.21

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Let the integer be $x$ . The situation gives $4x-6=3x+9$ . Subtracting $3x$ from both sides gives $x-6=9$ , so $x=15$ . Substitution verifies that both sides equal $54$ .

Q8. A student says, "The sum of two consecutive even integers is 46, so the integers must be 22 and 24 because they are close to half of 46." What is the best evaluation?

A.The reasoning is correct and complete

B.The numbers should be 21 and 25 because their sum is 46

C.The reasoning reaches the correct pair, but it should also verify that both numbers are consecutive even integers ✅

D.The numbers must be 23 and 23 because 46 divided by 2 is 23

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The proposed numbers $22$ and $24$ do satisfy the conditions, but merely being close to half the sum is not sufficient reasoning. A complete solution should show that they are even, differ by 2, and sum to 46.

Q9. A teacher records the possible value of $x$ against the difference between $x+12$ and $2x$ . A graph shows the difference equals zero at $x=12$ . What does this intersection represent?

A.The sum of the two expressions is 12

B.The two expressions have equal values when

x=12

✅

C.The smaller expression is always 12

D.The original number must be negative

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A zero difference means the two quantities are equal. At $x=12$ , the expressions $x+12$ and $2x$ both equal $24$ . Thus the graph identifies the value where the two modeled quantities are the same.

Q10. A number is divided by 3, and 4 is added to the result. The outcome is 10 less than the original number. Which equation should be solved, and what is the number?

x/3+4=x-10

, giving

x=21

✅

x/3-4=x-10

, giving

x=9

3x+4=x-10

, giving

x=-7

x/3+10=x-4

, giving

x=21

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The phrase divided by 3 gives $x/3$ , while adding 4 gives $x/3+4$ . Being 10 less than the original gives $x-10$ . Solving $x/3+4=x-10$ yields $x=21$ , which checks correctly.

Q11. Two numbers differ by 14, and their product is 240. Which pair satisfies both conditions?

A.10 and 24 ✅

B.12 and 26

C.8 and 22

D.14 and 28

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The pair must satisfy two conditions simultaneously: a difference of 14 and a product of 240. For $10$ and $24$ , the difference is $14$ and the product is $240$ , while the other choices fail at least one condition.

Q12. A student solves $3(x-4)=27$ by distributing to get $3x-4=27$ , then finds $x=31/3$ . Which correction explains the error?

A.The subtraction should occur after dividing, so

x=13

B.The 3 must multiply both

x

and

-4

, giving

3x-12=27

, so

x=13

✅

C.The equation should be

3x-4=9

, so

x=13/3

D.The negative sign means the equation has no solution

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The distributive property requires $3$ to multiply every term inside the parentheses. Thus $3(x-4)=3x-12$ , not $3x-4$ . Solving $3x-12=27$ gives $3x=39$ , so $x=13$ .

🔗 Related Topics (MCQs)

📝 How Use a Problem Solving Strategy MCQs 📝 How to Approach word problems MCQs 📝 How to Use a problem solving strategy for word problems MCQs 📝 Translate English sentences into algebraic equations MCQs 📝 Finding unknown numbers in algebra MCQs

←Previous Topic:How to Use a problem solving strategy for word problems📝 How to Solve number problemsTranslate English sentences into algebraic equations:Next Topic→