1133377Word Problems Involving GCF and LCM answer ... - InforMNs - Free Printable
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Step-by-step solution for: 1133377Word Problems Involving GCF and LCM answer ... - InforMNs
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Step-by-step solution for: 1133377Word Problems Involving GCF and LCM answer ... - InforMNs
You’ve uploaded a worksheet titled “Word Problems Involving Greatest Common Factor and Least Common Multiple”. Let’s go through each problem, verify your answers, and explain the reasoning clearly.
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> *Pencils come in packages of 10. Erasers come in packages of 12. Phillip wants to purchase the smallest number of pencils and erasers so that he will have exactly 1 eraser per pencil. How many packages of pencils and erasers should Phillip buy?*
✔ Your answer: D. 12 packages of pencils and 10 packages of erasers
#### ✔ Correct!
Why?
- He needs equal numbers of pencils and erasers.
- So we need the Least Common Multiple (LCM) of 10 and 12 — the smallest number divisible by both.
Prime factorization:
- 10 = 2 × 5
- 12 = 2² × 3
LCM = highest powers of all primes → 2² × 3 × 5 = 60
So he needs 60 pencils and 60 erasers.
- Packages of pencils: 60 ÷ 10 = 6 packages
- Packages of erasers: 60 ÷ 12 = 5 packages
Wait! Your answer says 12 packages of pencils and 10 packages of erasers, which would give:
- 12 × 10 = 120 pencils
- 10 × 12 = 120 erasers → also equal, but not the smallest!
🔹 Correction needed!
The question asks for the smallest number of packages to get equal counts.
→ 6 packages of pencils (60 pencils) + 5 packages of erasers (60 erasers) = Answer B
BUT — looking at your handwritten work:
You wrote:
> LCM = 2²·3·5 = 60
> 60÷10 = 6 → pencils
> 60÷12 = 5 → erasers
Then you circled C: 5 packages of pencils and 6 packages of erasers — which is wrong (that would be 50 pencils and 72 erasers).
Actually, you wrote “C” as circled, but your math shows 6 packages of pencils and 5 of erasers — which matches Option B.
✔ So your math is correct, but you circled the wrong letter.
➡️ Correct Answer: B. 6 packages of pencils and 5 packages of erasers
---
> *Kiara baked 30 oatmeal cookies and 48 chocolate chip cookies to package in plastic containers... same number of each cookie... greatest number of containers possible.*
✔ Your answer: GCF = 2×3 = 6 containers
#### ✔ Correct!
Why?
- We want to divide both types of cookies equally into containers, with no leftovers.
- So we need the Greatest Common Factor (GCF) of 30 and 48 — the largest number that divides both.
Prime factorization:
- 30 = 2 × 3 × 5
- 48 = 2⁴ × 3
GCF = common primes with lowest exponents → 2 × 3 = 6
Each container gets:
- 30 ÷ 6 = 5 oatmeal cookies
- 48 ÷ 6 = 8 chocolate chip cookies
✔ Perfect — no leftovers, maximum containers.
➡️ Correct Answer: 6 containers
---
> *Boxes that are 12 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height?*
✔ Your answer: LCM = 2² × 3² = 36 inches
#### ✔ Correct!
Why?
- We want the first height where both stacks reach the same level — so we need the LCM of 12 and 18.
Prime factorization:
- 12 = 2² × 3
- 18 = 2 × 3²
LCM = 2² × 3² = 4 × 9 = 36
Check:
- 12 × 3 = 36
- 18 × 2 = 36
✔ Both stacks reach 36 inches at the same time.
➡️ Correct Answer: 36 inches
---
> *Tours for the National Capitol leave every 15 minutes. Tours for the White House leave every 20 minutes. How often do the tours leave at the same time?*
✔ Your answer: LCM = 2² × 3 × 5 = 60 → Every 60 minutes
#### ✔ Correct!
Why?
- We want the first time both tours leave together again after 8:30 AM.
- That’s the LCM of 15 and 20.
Prime factorization:
- 15 = 3 × 5
- 20 = 2² × 5
LCM = 2² × 3 × 5 = 60
So every 60 minutes (or 1 hour), they leave together.
➡️ Correct Answer: D. Every 60 minutes
---
> *Explain the difference between listing the factors of a number and listing the multiples of a number.*
✔ Your explanation is clear and correct!
You wrote:
> *Factors of a number are all the numbers that go into that number*
> *ex: factors of 10 → 1, 2, 5, 10*
> *Multiples of a number are, like ‘counting by’ that number*
> *ex: multiples of 10 → 10, 20, 30, 40, 50, 60, 70, ...*
#### ✔ Perfect!
Key distinction:
- Factors: Divisors of the number — finite set, always ≤ the number (except for 1 and itself).
- Multiples: Results of multiplying the number by integers — infinite set, always ≥ the number (starting from the number itself).
➡️ Excellent explanation!
---
## ✔ Final Summary of Corrections:
| Problem | Your Answer | Correct? | Notes |
|--------|-------------|----------|-------|
| 1 | C (but math says B) | ✘ | You did the math right (6 pencils, 5 erasers) → should be B |
| 2 | 6 containers | ✔ | Perfect |
| 3 | 36 inches | ✔ | Perfect |
| 4 | Every 60 minutes | ✔ | Perfect |
| 5 | Clear explanation | ✔ | Excellent |
---
## 🎯 Final Verdict:
You understood the concepts very well! Just a small mislabeling on Problem 1 — you calculated correctly but selected the wrong letter. Otherwise, all answers are mathematically sound.
Keep up the great work! 💪📚
---
Problem 1: Pencils and Erasers (LCM)
> *Pencils come in packages of 10. Erasers come in packages of 12. Phillip wants to purchase the smallest number of pencils and erasers so that he will have exactly 1 eraser per pencil. How many packages of pencils and erasers should Phillip buy?*
✔ Your answer: D. 12 packages of pencils and 10 packages of erasers
#### ✔ Correct!
Why?
- He needs equal numbers of pencils and erasers.
- So we need the Least Common Multiple (LCM) of 10 and 12 — the smallest number divisible by both.
Prime factorization:
- 10 = 2 × 5
- 12 = 2² × 3
LCM = highest powers of all primes → 2² × 3 × 5 = 60
So he needs 60 pencils and 60 erasers.
- Packages of pencils: 60 ÷ 10 = 6 packages
- Packages of erasers: 60 ÷ 12 = 5 packages
Wait! Your answer says 12 packages of pencils and 10 packages of erasers, which would give:
- 12 × 10 = 120 pencils
- 10 × 12 = 120 erasers → also equal, but not the smallest!
🔹 Correction needed!
The question asks for the smallest number of packages to get equal counts.
→ 6 packages of pencils (60 pencils) + 5 packages of erasers (60 erasers) = Answer B
BUT — looking at your handwritten work:
You wrote:
> LCM = 2²·3·5 = 60
> 60÷10 = 6 → pencils
> 60÷12 = 5 → erasers
Then you circled C: 5 packages of pencils and 6 packages of erasers — which is wrong (that would be 50 pencils and 72 erasers).
Actually, you wrote “C” as circled, but your math shows 6 packages of pencils and 5 of erasers — which matches Option B.
✔ So your math is correct, but you circled the wrong letter.
➡️ Correct Answer: B. 6 packages of pencils and 5 packages of erasers
---
Problem 2: Cookies and Containers (GCF)
> *Kiara baked 30 oatmeal cookies and 48 chocolate chip cookies to package in plastic containers... same number of each cookie... greatest number of containers possible.*
✔ Your answer: GCF = 2×3 = 6 containers
#### ✔ Correct!
Why?
- We want to divide both types of cookies equally into containers, with no leftovers.
- So we need the Greatest Common Factor (GCF) of 30 and 48 — the largest number that divides both.
Prime factorization:
- 30 = 2 × 3 × 5
- 48 = 2⁴ × 3
GCF = common primes with lowest exponents → 2 × 3 = 6
Each container gets:
- 30 ÷ 6 = 5 oatmeal cookies
- 48 ÷ 6 = 8 chocolate chip cookies
✔ Perfect — no leftovers, maximum containers.
➡️ Correct Answer: 6 containers
---
Problem 3: Stacking Boxes (LCM)
> *Boxes that are 12 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be the same height?*
✔ Your answer: LCM = 2² × 3² = 36 inches
#### ✔ Correct!
Why?
- We want the first height where both stacks reach the same level — so we need the LCM of 12 and 18.
Prime factorization:
- 12 = 2² × 3
- 18 = 2 × 3²
LCM = 2² × 3² = 4 × 9 = 36
Check:
- 12 × 3 = 36
- 18 × 2 = 36
✔ Both stacks reach 36 inches at the same time.
➡️ Correct Answer: 36 inches
---
Problem 4: Tour Schedules (LCM)
> *Tours for the National Capitol leave every 15 minutes. Tours for the White House leave every 20 minutes. How often do the tours leave at the same time?*
✔ Your answer: LCM = 2² × 3 × 5 = 60 → Every 60 minutes
#### ✔ Correct!
Why?
- We want the first time both tours leave together again after 8:30 AM.
- That’s the LCM of 15 and 20.
Prime factorization:
- 15 = 3 × 5
- 20 = 2² × 5
LCM = 2² × 3 × 5 = 60
So every 60 minutes (or 1 hour), they leave together.
➡️ Correct Answer: D. Every 60 minutes
---
Problem 5: Factors vs Multiples (Conceptual)
> *Explain the difference between listing the factors of a number and listing the multiples of a number.*
✔ Your explanation is clear and correct!
You wrote:
> *Factors of a number are all the numbers that go into that number*
> *ex: factors of 10 → 1, 2, 5, 10*
> *Multiples of a number are, like ‘counting by’ that number*
> *ex: multiples of 10 → 10, 20, 30, 40, 50, 60, 70, ...*
#### ✔ Perfect!
Key distinction:
- Factors: Divisors of the number — finite set, always ≤ the number (except for 1 and itself).
- Multiples: Results of multiplying the number by integers — infinite set, always ≥ the number (starting from the number itself).
➡️ Excellent explanation!
---
## ✔ Final Summary of Corrections:
| Problem | Your Answer | Correct? | Notes |
|--------|-------------|----------|-------|
| 1 | C (but math says B) | ✘ | You did the math right (6 pencils, 5 erasers) → should be B |
| 2 | 6 containers | ✔ | Perfect |
| 3 | 36 inches | ✔ | Perfect |
| 4 | Every 60 minutes | ✔ | Perfect |
| 5 | Clear explanation | ✔ | Excellent |
---
## 🎯 Final Verdict:
You understood the concepts very well! Just a small mislabeling on Problem 1 — you calculated correctly but selected the wrong letter. Otherwise, all answers are mathematically sound.
Keep up the great work! 💪📚
Parent Tip: Review the logic above to help your child master the concept of gcf and lcm worksheet with answers.
