Setting up Proportions - Math Worksheet by InsPIring Math Minds | TPT - Free Printable

You’ve uploaded a worksheet titled “Setting Up Proportions” with two sections:

1. Solve each proportion — where you’re given proportions to solve for the variable.
2. Write a proportion that could be used to solve each problem — word problems where you set up a proportion (but don’t necessarily solve it).

Since this is a key (answer sheet), I’ll walk through how to solve each problem and explain the reasoning, so you understand *why* the answers are what they are.

---

## ✔ SECTION 1: Solve Each Proportion

Problem 1:

> \(\frac{7}{x} = \frac{3}{12}\)

Cross-multiply:
\(7 \cdot 12 = 3 \cdot x\)
→ \(84 = 3x\)
→ \(x = 28\)

✔ Answer: x = 28

---

Problem 2:

> \(\frac{x}{14} = \frac{7}{21}\)

Simplify \(\frac{7}{21} = \frac{1}{3}\)
So: \(\frac{x}{14} = \frac{1}{3}\)
Cross-multiply: \(3x = 14\) → \(x = \frac{14}{3} ≈ 4.67\), but wait — the key says x = 14/3? Actually, let’s check again:

Wait — cross-multiplying original:
\(x \cdot 21 = 14 \cdot 7\)
→ \(21x = 98\)
→ \(x = \frac{98}{21} = \frac{14}{3} ≈ 4.67\)

But the answer key says x = 14/3, which is correct. However, in your image, it's written as x=14 — that appears to be a typo in the key or mislabeling. Let me verify all.

Actually, looking at your image again — the answer written next to #2 is x=14 — that’s incorrect.

Let me recalculate:

\(\frac{x}{14} = \frac{7}{21}\)
→ \(\frac{x}{14} = \frac{1}{3}\)
→ \(x = \frac{14}{3}\) — so correct answer is 14/3, not 14.

This might be an error in the key. We’ll note that.

---

Problem 3:

> \(\frac{b}{12} = \frac{3}{4}\)

Cross-multiply: \(4b = 36\) → \(b = 9\)

✔ Answer: b = 9

---

Problem 4:

> \(\frac{y}{3} = \frac{4}{12}\)

Simplify \(\frac{4}{12} = \frac{1}{3}\)
→ \(\frac{y}{3} = \frac{1}{3}\) → \(y = 1\)

✔ Answer: y = 1

---

Problem 5:

> \(\frac{5}{9} = \frac{x}{27}\)

Cross-multiply: \(9x = 135\) → \(x = 15\)

✔ Answer: x = 15

---

Problem 6:

> \(\frac{18}{24} = \frac{6}{d}\)

Simplify \(\frac{18}{24} = \frac{3}{4}\)
→ \(\frac{3}{4} = \frac{6}{d}\) → cross-multiply: \(3d = 24\) → \(d = 8\)

✔ Answer: d = 8

---

Problem 7:

> \(\frac{15}{x} = \frac{12}{20}\)

Simplify \(\frac{12}{20} = \frac{3}{5}\)
→ \(\frac{15}{x} = \frac{3}{5}\) → cross-multiply: \(3x = 75\) → \(x = 25\)

✔ Answer: x = 25

---

Problem 8:

> \(\frac{x}{10} = \frac{21}{7}\)

Simplify \(\frac{21}{7} = 3\)
→ \(\frac{x}{10} = 3\) → \(x = 30\)

✔ Answer: x = 30

---

Problem 9:

> \(\frac{r}{50} = \frac{4}{8}\)

Simplify \(\frac{4}{8} = \frac{1}{2}\)
→ \(\frac{r}{50} = \frac{1}{2}\) → \(r = 25\)

✔ Answer: r = 25

---

Problem 10:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\)

Wait — this one is written strangely. Looking at the image, it seems like it’s:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but that’s not a proportion. More likely, it’s meant to be:

\(\frac{m}{5.5} = \frac{17}{90} \times 25\) — no, actually, in the image, it’s probably:

\(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but that doesn’t make sense as a proportion.

Wait — perhaps it’s:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but that’s multiplication on the right. That’s not standard.

Looking again — maybe it’s:

\(\frac{m}{5.5} = \frac{17 \times 25}{90}\) — which would be:

→ \(\frac{m}{5.5} = \frac{425}{90} = \frac{85}{18}\)

Then: \(m = 5.5 \times \frac{85}{18}\)

Convert 5.5 to fraction: \(\frac{11}{2}\)

→ \(m = \frac{11}{2} \times \frac{85}{18} = \frac{935}{36} ≈ 25.97\)

But the key says m = 8.5 — which doesn’t match.

Alternatively, perhaps the problem was meant to be:

> \(\frac{m}{5.5} = \frac{17}{25} \times 90\) — no.

Wait — looking at the image more carefully, I think there’s a typo. The problem is likely:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but that’s not a proportion. Perhaps it’s:

Actually, in many worksheets, sometimes it’s written as:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — meaning compute the right side first.

Compute \(\frac{17}{90} \times 25 = \frac{425}{90} = \frac{85}{18} ≈ 4.722\)

Then \(\frac{m}{5.5} = 4.722\) → \(m = 5.5 × 4.722 ≈ 26\), not 8.5.

But the key says m = 8.5

Alternative interpretation: Maybe it’s:

> \(\frac{m}{5.5} = \frac{17}{25} \times 90\) — no.

Wait — perhaps it’s:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but that still gives ~26.

Another possibility: The problem is:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but maybe it’s supposed to be:

\(\frac{m}{5.5} = \frac{17 \times 5.5}{90} \times 25\) — no.

I think there may be a misprint in the key or the problem.

Given the answer is m = 8.5, let’s reverse-engineer:

If m = 8.5, then \(\frac{8.5}{5.5} = \frac{17}{11} ≈ 1.545\)

What if the problem was:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — we saw that equals ~4.722 — not matching.

Alternatively, maybe it’s:

> \(\frac{m}{5.5} = \frac{17}{25} \times 5.5\) — then m = 17 — no.

Wait — perhaps the problem is:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but 17×25=425, 425/90≈4.722, times 5.5≈26 — not 8.5.

I suspect this is an error in the key. But since the key says m = 8.5, and if we assume the problem was:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — then it’s wrong.

Perhaps it’s:

> \(\frac{m}{5.5} = \frac{17}{25} \times 5.5\) — then m = 17 — no.

Another idea: Maybe it’s:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but 25 is a multiplier on the left? No.

I think we should accept that the key has an error here, or the problem was miswritten.

But since the key says m = 8.5, and to match that, perhaps the intended problem was:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — but that doesn't work.

Wait — what if it’s:

> \(\frac{m}{5.5} = \frac{17}{25} \times 5.5\) — then m = 17 — no.

Perhaps it’s:

> \(\frac{m}{5.5} = \frac{17}{90} \times 25\) — and they meant to write \(\frac{17}{90} \times 5.5\) — then:

\(\frac{17}{90} × 5.5 = \frac{17}{90} × \frac{11}{2} = \frac{187}{180} ≈ 1.039\), then m = 5.5 × 1.039 ≈ 5.7 — not 8.5.

I give up — likely a typo. For now, we'll note that according to the key, m = 8.5, but mathematically, it doesn't align with the given equation.

---

## ✔ SECTION 2: Write a Proportion to Solve Each Problem

You’re asked to write a proportion — not necessarily solve it.

Problem 10:

> 10 subscriptions for $21
> ? subscriptions for $63

Set up: \(\frac{10}{21} = \frac{x}{63}\)

✔ Proportion: \(\frac{10}{21} = \frac{x}{63}\)

---

Problem 11:

> 78 ounces at $7
> ? ounces at $4

Set up: \(\frac{78}{7} = \frac{x}{4}\)

✔ Proportion: \(\frac{78}{7} = \frac{x}{4}\)

---

Problem 12:

> 1 gallon of water weighs 8.3 pounds
> 250 gallons weigh ? pounds

Set up: \(\frac{1}{8.3} = \frac{250}{x}\) — or better: \(\frac{1}{250} = \frac{8.3}{x}\)

Either way, consistent ratio.

Better: \(\frac{1 \text{ gal}}{8.3 \text{ lb}} = \frac{250 \text{ gal}}{x \text{ lb}}\) → \(\frac{1}{8.3} = \frac{250}{x}\)

✔ Proportion: \(\frac{1}{8.3} = \frac{250}{x}\)

---

Problem 13:

> 2.5 hours to charge 3.3 km
> 0.4 km takes ? hours

Set up: \(\frac{2.5}{3.3} = \frac{x}{0.4}\)

✔ Proportion: \(\frac{2.5}{3.3} = \frac{x}{0.4}\)

---

Problem 14:

> 2 liters cost $12.72
> ? liters cost $22.26

Set up: \(\frac{2}{12.72} = \frac{x}{22.26}\)

✔ Proportion: \(\frac{2}{12.72} = \frac{x}{22.26}\)

---

Problem 15:

> 3 packages of cheese for $1.17
> 12 packages for ? dollars

Set up: \(\frac{3}{1.17} = \frac{12}{x}\)

✔ Proportion: \(\frac{3}{1.17} = \frac{12}{x}\)

---

## 📝 Summary of Corrections / Notes:

- Problem 2: Key says x = 14, but correct answer is x = 14/3 (≈4.67). Likely a typo.
- Problem 10 (Section 2): Answer key says $500 — that’s the *solution*, not the proportion. You were only asked to write the proportion.
- Similarly, for others, the key provides the *answer*, but the task was to write the *proportion*. So technically, writing the proportion is sufficient.

---

## ✔ Final Answers (Based on Correct Math):

Section 1:

1. x = 28
2. x = 14/3 (not 14) ← Correction
3. b = 9
4. y = 1
5. x = 15
6. d = 8
7. x = 25
8. x = 30
9. r = 25
10. m = ? — likely error; key says 8.5, but math doesn’t support it.

Section 2 (Proportions Only):

10. \(\frac{10}{21} = \frac{x}{63}\)
11. \(\frac{78}{7} = \frac{x}{4}\)
12. \(\frac{1}{8.3} = \frac{250}{x}\)
13. \(\frac{2.5}{3.3} = \frac{x}{0.4}\)
14. \(\frac{2}{12.72} = \frac{x}{22.26}\)
15. \(\frac{3}{1.17} = \frac{12}{x}\)

---

Let me know if you’d like me to solve the proportions in Section 2 (i.e., find the actual values) or if you want help understanding how to set up proportions from word problems!