Worksheet: Solving Proportions - Using Ratios | Pre-Algebra Printable - Free Printable

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Let’s go through each problem one by one and explain how to solve them using proportions.

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Problem 1:

> "Tobias's Math 'Taser' will 'Taser' him every 9 seconds. If it has already Tased him 43 times, how many times will it have Tased him after 2 minutes?"

#### Step-by-step Solution:

- We are told that the taser goes off every 9 seconds, so:
- In 9 seconds → 1 tase
- So, in 1 second → $ \frac{1}{9} $ tases

- Total time = 2 minutes = $ 2 \times 60 = 120 $ seconds

- Number of tases in 120 seconds:
$$
\frac{120}{9} = 13.\overline{3} \approx 13.33
$$

But since we're asked how many times it will have Tased him, and it only Tases at full intervals, we take the whole number part:
→ 13 times

Wait — but the answer key says 81 times, which suggests a misunderstanding.

Wait! Let's check the answer key again:

> Answer: 81 times

That doesn't match our calculation.

But let’s look again — maybe the "already Tased him 43 times" is just context? But the question says:

> "If it has already Tased him 43 times, how many times will it have Tased him after 2 minutes?"

Ah — this is ambiguous. But the answer key says 81, and the setup shows:

> $ \frac{1}{9} = \frac{x}{729} $

Wait — let’s look at the solution shown in the key:

> $ \frac{1}{9} = \frac{x}{729} $ → $ x = 81 $

So they’re saying:
In 9 seconds → 1 tase
In 729 seconds → ? tases

So total time = 729 seconds?

But 729 seconds = $ \frac{729}{60} = 12.15 $ minutes — not 2 minutes.

Wait — something is wrong here.

Let me re-read the problem carefully.

Actually, looking back:

> "Tobias's Math 'Taser' will 'Taser' him every 9 seconds. If it has already Tased him 43 times, how many times will it have Tased him after 2 minutes?"

The answer key says 81, but 2 minutes = 120 seconds.

Number of 9-second intervals in 120 seconds:
$ \frac{120}{9} = 13.\overline{3} $ → 13 full intervals → 13 more tases.

So total tases = 43 + 13 = 56

But answer key says 81 — so clearly, either the problem or the key is misaligned.

Wait — perhaps the "after 2 minutes" means in total over 2 minutes, not in addition to the 43?

But then:
In 120 seconds → $ \frac{120}{9} = 13.\overline{3} $ → 13 tases (if counting only complete cycles)

But 13 ≠ 81.

Alternatively, maybe the "43 times" was just an example, and the real question is:

> "How many times will it Tase him in 2 minutes?"

Then:
$ \frac{1}{9} = \frac{x}{120} $ → $ x = \frac{120}{9} = 13.\overline{3} $ → 13 times

Still not 81.

But wait — the key says:
> $ \frac{1}{9} = \frac{x}{729} $ → $ x = 81 $

So 729 seconds = 12.15 minutes

So if the taser fires every 9 seconds, in 729 seconds it fires $ \frac{729}{9} = 81 $ times.

So the time must be 729 seconds, not 2 minutes.

But 2 minutes is 120 seconds.

So unless there's a typo, this suggests the original problem may have said "12 minutes" or "729 seconds", not "2 minutes".

Possibility: The original problem might have been:

> "If it has already Tased him 43 times, how many times will it have Tased him after 12 minutes?"

12 minutes = 720 seconds → $ \frac{720}{9} = 80 $ tases → total = 43 + 80 = 123

Still not 81.

Alternatively, maybe the "43 times" is irrelevant, and the question is:
"How many times does it fire in 729 seconds?" → $ \frac{729}{9} = 81 $

So likely, the original problem was:

> "Tobias's Math 'Taser' fires every 9 seconds. How many times will it fire in 729 seconds?"

Then:
$$
\frac{1}{9} = \frac{x}{729} \Rightarrow x = \frac{729}{9} = 81
$$

So the answer is 81 times.

Thus, the problem likely had a typo — it should be 729 seconds, not 2 minutes.

✔ Conclusion: The proportion is set up correctly as $ \frac{1}{9} = \frac{x}{729} $, leading to $ x = 81 $. But the time should be 729 seconds (12.15 minutes), not 2 minutes.

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Problem 2:

> "The ratio of apples to pears in a basket is 3:4. If there are 12 pears, how many apples are there?"

#### Solution:

Given:
Apples : Pears = 3 : 4
Pears = 12

Set up proportion:
$$
\frac{3}{4} = \frac{x}{12}
$$

Cross-multiply:
$$
3 \times 12 = 4x \Rightarrow 36 = 4x \Rightarrow x = 9
$$

✔ Answer: 9 apples

This matches the key.

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Problem 3:

> "54 pounds of jujy gummy bears costs $450. How many pounds of bears can you get for $30?"

#### Solution:

Set up proportion:
$$
\frac{54}{450} = \frac{x}{30}
$$

Cross-multiply:
$$
54 \times 30 = 450x \Rightarrow 1620 = 450x \Rightarrow x = \frac{1620}{450} = 3.6
$$

✔ Answer: 3.6 pounds

Key says: $ \frac{54}{450} = \frac{x}{30} $ → $ x = 3.6 $ → correct.

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Problem 4:

> "If the average teenager complains about 13 things every four minutes, how many complaints will he make in 16 minutes?"

#### Solution:

Set up proportion:
$$
\frac{13}{4} = \frac{x}{16}
$$

Cross-multiply:
$$
13 \times 16 = 4x \Rightarrow 208 = 4x \Rightarrow x = 52
$$

✔ Answer: 52 complaints

Matches key.

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Problem 5:

> "The ratio of students with two siblings to students with one sibling is 48 to 23. If there are 71 students with one sibling, how many have two siblings?"

#### Solution:

Set up proportion:
$$
\frac{48}{23} = \frac{x}{71}
$$

Cross-multiply:
$$
48 \times 71 = 23x \Rightarrow 3408 = 23x \Rightarrow x = \frac{3408}{23} = 148.17...
$$

Wait — but the key says 148.

Let’s calculate:
$$
23 \times 148 = 23 \times 150 - 23 \times 2 = 3450 - 46 = 3404
$$

3408 - 3404 = 4 → not exact.

Wait — maybe the ratio is reversed?

Wait — the ratio is:
> "students with two siblings : students with one sibling = 48 : 23"

So:
$$
\frac{48}{23} = \frac{x}{71}
\Rightarrow x = \frac{48 \times 71}{23} = \frac{3408}{23} = 148.17...
$$

Not a whole number.

But the key says 148.

Wait — maybe it's a rounding error?

Or maybe the numbers are off.

But let’s check:
If 23 parts = 71 students → 1 part = $ \frac{71}{23} \approx 3.087 $

Then 48 parts = $ 48 \times 3.087 \approx 148.17 $

So not exactly 148.

But the key says 148 — so either:
- The problem meant 70 students with one sibling? Or
- The ratio is approximate?

Alternatively, maybe the key is wrong.

Wait — the key says:
> $ \frac{48}{23} = \frac{x}{71} $ → $ x = 148 $

But $ \frac{48}{23} \times 71 = \frac{3408}{23} = 148.173... $

So not exactly 148.

But perhaps it's intended to be rounded down.

Alternatively, maybe the problem says "approximately"?

But the key gives 148, so we’ll accept it as given.

✔ Answer: 148 students with two siblings

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Tricky Problem:

> "The ratio of boys to girls in the sixth grade is 3:5. There are 72 students. How many STUDENTS are boys?"

#### Solution:

Ratio: Boys : Girls = 3 : 5
Total parts = 3 + 5 = 8

Total students = 72

So:
- Each part = $ \frac{72}{8} = 9 $
- Boys = 3 parts = $ 3 \times 9 = 27 $
- Girls = 5 parts = $ 5 \times 9 = 45 $

✔ Answer: 27 boys

Key says: $ \frac{3}{8} = \frac{x}{72} \Rightarrow x = 27 $ → correct.

Note: The key says this is a tricky proportion because it involves total parts, not direct comparison.

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✔ Summary of All Answers:

| Problem | Answer | Explanation |
|--------|--------|-----------|
| 1 | 81 times | Proportion: $ \frac{1}{9} = \frac{x}{729} $ → $ x = 81 $; likely time was 729 seconds, not 2 minutes |
| 2 | 9 apples | $ \frac{3}{4} = \frac{x}{12} $ → $ x = 9 $ |
| 3 | 3.6 pounds | $ \frac{54}{450} = \frac{x}{30} $ → $ x = 3.6 $ |
| 4 | 52 complaints | $ \frac{13}{4} = \frac{x}{16} $ → $ x = 52 $ |
| 5 | 148 students | $ \frac{48}{23} = \frac{x}{71} $ → $ x \approx 148.17 $ → rounded to 148 |
| Tricky | 27 boys | Ratio 3:5 → total 8 parts → $ \frac{3}{8} \times 72 = 27 $ |

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⚠️ Important Note:

There appears to be a discrepancy in Problem 1 — the time stated (2 minutes) does not align with the answer (81 times). The correct time for 81 tases at 9-second intervals is:
$$
81 \times 9 = 729 \text{ seconds} = 12.15 \text{ minutes}
$$

So unless the original problem said 729 seconds, the problem statement is incorrect.

Similarly, Problem 5 has a non-integer result, but the key rounds it to 148.

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