Dimensional Analysis Practice worksheet - Free Printable

Let's solve each problem step-by-step using dimensional analysis (also known as the factor-label method). We'll use the given relationships and show all conversion steps.

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Problem 1:

A runner competed in a 5-mile run. How many yards did she run?

Given relationship:
1 mile = 1760 yards

Set up the conversion:

$$
5 \text{ miles} \times \frac{1760 \text{ yds}}{1 \text{ mile}} = ?
$$

Now calculate:

$$
5 \times 1760 = 8,800 \text{ yards}
$$

✔ Answer: 8,800 yards

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Problem 2:

In the Tour de France, cyclists ride 3,653.6 km over 20 days. How many miles do they go? [Hint: watch for unimportant information!]

We are asked to convert kilometers to miles, so the 20 days is irrelevant (just a distractor).

Given relationship:
1 mile = 1.61 km → So, $ \frac{1 \text{ mi}}{1.61 \text{ km}} $

Set up the conversion:

$$
3,653.6 \text{ km} \times \frac{1 \text{ mi}}{1.61 \text{ km}} = ?
$$

Now calculate:

$$
\frac{3,653.6}{1.61} \approx 2,270 \text{ miles}
$$

(Using calculator: $ 3,653.6 \div 1.61 = 2,270 $)

✔ Answer: 2,270 miles

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Problem 3:

After a nice meal, perhaps you’d finish it off with a pound cake for dessert. What would the name of this cake be in grams?

Given relationships:
- 1 lb = 16 oz
- 1 oz = 28.35 g

So we need to convert 1 lb → grams via ounces.

Set up the conversion:

$$
1 \text{ lb} \times \frac{16 \text{ oz}}{1 \text{ lb}} \times \frac{28.35 \text{ g}}{1 \text{ oz}} = ?
$$

Now calculate:

First: $ 1 \times 16 = 16 $ oz
Then: $ 16 \times 28.35 = 453.6 $ g

✔ Answer: 453.6 grams → A "453.6-g cake"

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Problem 4:

How many liters of milk would you need to equal one gallon?

Given relationships:
- 1 L = 1.06 qts
- 4 qts = 1 gal

We want to convert 1 gallon → liters, so we go through quarts.

Set up the conversion:

$$
1 \text{ gal} \times \frac{4 \text{ qts}}{1 \text{ gal}} \times \frac{1 \text{ L}}{1.06 \text{ qts}} = ?
$$

Now calculate:

First: $ 1 \times 4 = 4 $ qts
Then: $ 4 \div 1.06 \approx 3.7736 $ L

Round to reasonable precision: ≈ 3.77 L

✔ Answer: Approximately 3.77 liters

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Problem 5:

If you go to school for 180 days each year and each day is 7 hours long, how many hours are spent in school in one year?

Given:
- 180 days per school year
- 7 hours per school day

We just multiply:

$$
180 \text{ days/year} \times 7 \text{ hrs/day} = 1,260 \text{ hours/year}
$$

We can write it using dimensional analysis:

$$
\frac{180 \text{ days}}{1 \text{ (school) year}} \times \frac{7 \text{ hrs}}{1 \text{ day}} = 1,260 \text{ hrs/year}
$$

✔ Answer: 1,260 hours per school year

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | 8,800 yards |
| 2 | 2,270 miles |
| 3 | 453.6 grams |
| 4 | ~3.77 liters |
| 5 | 1,260 hours |

All conversions used proper dimensional analysis with correct unit cancellation. Let me know if you'd like these formatted as boxed answers or printed!