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📝 Clear decimals by multiplying by the LCD (13 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 13 questions available

←Previous Topic:Clear fractions by multiplying by the LCD📝 Clear decimals by multiplying by the LCDHow to Solve a Formula for a Specific Variable:Next Topic→

What is Clear decimals by multiplying by the LCD?

Definition:
Clearing decimals by multiplying by the LCD (or a power of 10) means multiplying both sides of the equation by $10^n$ , where $n$ is the greatest number of decimal places in any term. This converts all decimals to integers, simplifying the equation.

Working:
For $0.3x + 0.02 = 0.15$ , the max decimal places is 2 (in 0.02), so multiply by $10^2 = 100$ : $100(0.3x) + 100(0.02) = 100(0.15)$ , giving $30x + 2 = 15$ . Solve: $30x = 13$ , $x = 13/30$ .

Example:
Solve $0.25a - 0.1 = 0.4$ . Max decimal places = 2 (0.25, 0.10, 0.40). Multiply by 100: $25a - 10 = 40$ , add 10: $25a = 50$ , $a = 2$ .

Reason:
This removes decimal points, reducing the chance of place-value errors and making the equation more approachable.

Easy

Medium

Hard

📝 All Clear decimals by multiplying by the LCD MCQs

Q1. Which value of $x$ satisfies $0.4x+1.2=2.8$ ?

A.2

B.4 ✅

C.6

D.7

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying the entire equation by 10 clears the decimals and gives $4x+12=28$ . Subtracting 12 produces $4x=16$ , so $x=4$ . The key idea is multiplying every term by the same factor.

Q2. What is the main reason multiplying every term of a decimal equation by the least common denominator helps?

A.It changes the solution of the equation

B.It removes decimal places while preserving equality ✅

C.It makes only the variable an integer

D.It eliminates the need to perform inverse operations

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The multiplier is chosen so that every decimal becomes an integer. Because the same nonzero factor multiplies both sides, equality and therefore the solution set remain unchanged.

Q3. A student multiplies $0.25x+0.6=1.1$ by 100 and writes $25x+6=110$ . What should be corrected?

A.Nothing; the equation is correct

B.Only

0.25x

should be multiplied by 100

C.The right side should remain

1.1

D.The

0.6

term should become 60, so the correct equation is

25x+60=110

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Multiplying by 100 affects every term. Thus $0.25x$ becomes $25x$ , $0.6$ becomes 60, and $1.1$ becomes 110. The student's error comes from treating terms inconsistently.

Q4. Which multiplier most efficiently clears all decimals in $0.125x+0.75=2.5$ ?

A.10

B.20

C.100

D.1000 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The decimal $0.125$ has three decimal places, so multiplying by 1000 converts every decimal in the equation into an integer. Smaller powers of 10 would leave at least one decimal.

Q5. A shop charges $0.75x+2.50=8.50$ , where $x$ is the number of identical items. Which equation results after clearing decimals using 100?

75x+250=850

✅

7.5x+25=85

75x+25=85

750x+250=850

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Multiplying the complete equation by 100 converts $0.75$ to 75, $2.50$ to 250, and $8.50$ to 850. The resulting equation is $75x+250=850$ , which can then be solved.

Q6. Consider $0.06x+0.4=1.12$ . Two students use different multipliers. Student A multiplies by 100, while Student B multiplies by 1000. Which conclusion is most accurate?

A.Only Student A can obtain the correct solution

B.Only Student B can obtain the correct solution

C.Both can obtain the same solution if they multiply every term correctly ✅

D.The two multipliers necessarily produce different solutions

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both 100 and 1000 are valid nonzero multipliers because each clears all decimals. Student B creates larger integers, but the equation remains equivalent. Correctly applying the multiplier to every term leads to the same solution.

Q7. A recipe uses $0.35x+1.2=4.7$ , where $x$ represents the number of identical batches. After clearing decimals with 100, which solution does the resulting equation produce?

A.8

B.10 ✅

C.12

D.14

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Multiplying by 100 gives $35x+120=470$ . Subtracting 120 gives $35x=350$ , and dividing by 35 gives $x=10$ . The result also makes practical sense because the number of batches is a whole number.

Q8. A student solves $0.8x-1.6=4.8$ by multiplying only the terms containing decimals, obtaining $8x-16=4.8$ . Why is this reasoning invalid?

A.The multiplier 10 is too large

B.The variable cannot be multiplied by 10

C.Every term on both sides must be multiplied by the same factor ✅

D.Subtraction cannot be used after clearing decimals

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: An equation stays equivalent only when the same nonzero multiplier is applied to every term on both sides. Leaving $4.8$ unchanged changes the relationship and can produce an incorrect solution.

Q9. A student claims that $0.3x+0.9=2.1$ becomes $3x+9=2.1$ after multiplying by 10. Which statement best evaluates the work?

A.Correct, because only the left side contains decimals

B.Incorrect, because

2.1

must also become 21 ✅

C.Correct, because constants do not need multiplication

D.Incorrect, because

0.3

should become 30

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Multiplying by 10 must apply to the entire equation. Therefore $0.3x$ becomes $3x$ , $0.9$ becomes 9, and $2.1$ becomes 21. Omitting the right-side multiplication destroys equivalence.

Q10. A number-line graph represents the equation $0.5x+1.5=4$ with a single marked solution. Which location should be marked?

x=3

x=4

x=5

✅

x=6

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Multiplying by 2 gives $x+3=8$ , so $x=5$ , not 3. Therefore the correct graph location is 5. Checking directly, $0.5(5)+1.5=2.5+1.5=4$ , confirming the solution.

Q11. Two methods solve $0.2x+0.6=2.4$ . Method A multiplies by 10 first; Method B multiplies by 100 first. Which comparison is correct?

A.Method A is invalid because 10 is not large enough

B.Method B is invalid because multipliers must be less than 10

C.Both methods preserve the solution, but Method A usually involves simpler arithmetic ✅

D.Only Method B preserves equality

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Both 10 and 100 are valid nonzero multipliers, and each clears the decimals. However, multiplying by 10 produces $2x+6=24$ , which is simpler than the larger numbers produced by multiplying by 100.

Q12. A rectangular garden has length $1.5x+2$ meters and a fixed width of 4 meters. Its perimeter is 20 meters. Which equation and solution correctly model the situation?

1.5x+2+4=20, x=9.33

2(1.5x+2+4)=20, x=2

✅

1.5x+2(4)=20, x=8

2(1.5x+2)+4=20, x=6.67

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: For a rectangle, perimeter equals twice the sum of length and width, so $2(1.5x+2+4)=20$ . Dividing by 2 gives $1.5x+6=10$ , hence $1.5x=4$ and $x=2$ .

Q13. For $0.125x+0.375=1.625$ , a student argues that multiplying by 8 is enough because 8 is the denominator of $0.125$ . Is the argument valid, and what is $x$ ?

A.Yes;

x=10

✅

B.No; 8 does not clear every decimal, and

x=10

C.Yes;

x=12

D.No; the equation has no solution

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Multiplying by 8 changes the equation to $x+3=13$ , so it actually does clear these decimals because $0.375=3/8$ and $1.625=13/8$ . Thus the argument that 8 is insufficient is false, while $x=10$ is correct. Since option B says both that 8 does not clear every decimal and $x=10$ , it is internally inconsistent. The correct conclusion is therefore A.

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

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