Lesson 3.9Equivalent Ratios
Start hereWords, worked example, and sentence starters
Learning target I can find and check equivalent ratios using multiplication and division.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| Equivalent RatiosSpanish: Razones equivalentes | Ratios that compare quantities in the same relationship. | 2:3 = 4:6 because both parts were multiplied by 2. |
| RatioSpanish: Razón | A way to compare two amounts, like 3 to 2. | 3 red apples : 5 green apples |
| RateSpanish: Tasa | A ratio comparing two amounts with different units, like miles per hour. | 60 miles per 1 hour |
| ProportionSpanish: Proporción | A math sentence saying two ratios are equal. | 2/3 = 4/6 |
| SimplifySpanish: Simplificar | To write a ratio with smaller numbers that still compares the same way. 6:8 and 3:4 are the same ratio. | 12:18 → divide both by 6 → 2:3 |
| Ratio tableSpanish: Tabla de razones | A table of equal ratios that shows a pattern. | Cups of juice: 2, 4, 6 | Cups of water: 3, 6, 9 |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Chef Montoya's soup serves 8 people, but she needs to serve 32.
- I find the scale factor: 32 divided by 8 = 4.
- The recipe uses 3 cups of tomatoes to 2 cups of broth, a 3:2 ratio.
- I multiply both by 4: 3 x 4 = 12 cups of tomatoes, and 2 x 4 = 8 cups of broth.
- So 3:2 and 12:8 are equivalent ratios, and the flavor stays the same.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
A ratio is 2:5. We want to make it bigger using a scale factor of 3.
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- I multiplied and by the same number, , so the ratio is equivalent to 3:2.
Word bank Equivalent RatiosRatioRateProportionSimplifyRatio tablemultiplysame numberequivalentscaletomatoesbroth
Watch outA common mistake
A common mistake in Equivalent Ratios is ADDING the scale factor to one part instead of MULTIPLYING both parts by it. For example, scaling 2:5 by a factor of 4 should give 8:20 (2×4 and 5×4) — not 6:9 (2+4 and 5+4). Adding changes the taste of the recipe (and the value of the ratio); only multiplying or dividing both numbers by the same amount keeps it equivalent.
Lesson 3.9Equivalent Ratios
Version ASupported practice
Learning target I can find and check equivalent ratios using multiplication and division.
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1Write the name of the correct group on each line.
Sort each pair of ratios: are they equivalent or not equivalent?
- 2:4 and 3:6
- 5:10 and 1:2
- 3:7 and 6:12
- 4:9 and 8:16
Hint Two ratios are equivalent if one is the other with BOTH parts scaled by the same multiplier.
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2Circle the letter of the best answer. Show how you know.
Which ratio is equivalent to 2:5?
- A2:10
- B3:6
- C4:10
- D5:2
Hint Equivalent ratios come from multiplying BOTH parts of 2:5 by the same number.
Show your work -
3Circle the letter of the best answer. Show how you know.
A recipe uses 4 eggs for every 6 cups of flour. Which ratio is equivalent?
- A2:3
- B4:3
- C6:4
- D8:10
Hint You can scale a ratio DOWN too — divide both parts by the same number.
Show your work -
4Circle the letter of the best answer. Show how you know.
Which ratio is NOT equivalent to 6:9?
- A2:3
- B3:5
- C12:18
- D18:27
Hint First simplify 6:9 to its simplest form — that becomes your measuring stick.
Show your work -
5Circle the letter of the best answer. Show how you know.
What is the simplest form of the ratio 15:25?
- A1:5
- B3:5
- C5:3
- D15:25
Hint Simplest form means both numbers have been divided by their greatest common factor.
Show your work
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6Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake — Lemonade Ratio
- 1Set up the ratio2 cups lemon : 5 cups water = 6 cups lemon : ? cups water
- 2Find the multiplier6 ÷ 2 = 3
- 3Apply to the water5 + 3 = 8
- 4Answer6 cups lemon : 8 cups water
Which step has the mistake? Explain what went wrong, then show the correct work.
Hint Look at Step 3. The lemon was multiplied by 3, so what should you do to the water?
Correct workCorrect answer:
Explain your thinkingHow did you find the missing values in Chef Montoya's table? How can you PROVE two of your rows are equivalent ratios?
Sentence starter I multiplied ___ and ___ by the same number, ___, so the ratio ___ is equivalent to 3:2.
Lesson 3.9Equivalent Ratios
Version BCore practice
Learning target I can find and check equivalent ratios using multiplication and division.
-
1Write the name of the correct group on each line.
Sort these ratios — which are equivalent to 3:4 and which are not?
- 6:8
- 9:12
- 15:20
- 3:5
- 4:3
- 6:12
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2Complete the table. Show how you found each value.
Find the missing value that makes each pair of ratios equivalent.
Ratio 1 Ratio 2 Missing Value 3:7 9:? 5:8 ?:24 4:6 ?:3 Show your work -
3Use the model to solve. Show your work.
Each part = 25 ÷ 5 = 5 cups. Juice = 3 × 5 = 15 cups. Water = 2 × 5 = 10 cups.
Show your workAnswer:
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4Circle the letter of the best answer. Show how you know.
Marcus earns $45 for every 3 hours of work. How much does he earn in 8 hours?
- A$90
- B$105
- C$120
- D$135
Show your work
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5Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake — Simplifying a Ratio
- 1Set up the ratioA student wants to write 12:18 in simplest form.
- 2Find the common factorBoth 12 and 18 can be divided by 6.
- 3Divide each part12 ÷ 6 = 2 and 18 ÷ 3 = 6, so they wrote 2:6.
- 4Answer12:18 simplifies to 2:6
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer:
Explain your thinkingHow did you find the missing values in Chef Montoya's table? How can you PROVE two of your rows are equivalent ratios?
Lesson 3.9Equivalent Ratios
ChallengeExtension
Learning target I can find and check equivalent ratios using multiplication and division.
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1Complete the table. Show how you found each value.
A student claims these ratios are all equivalent. Check each one and mark whether it simplifies to 2:5 or not.
Ratio Simplified Form Equivalent to 2:5? 4:10 2:5 Yes 6:15 8:25 10:25 Show your work
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2Find the mistake. Explain it, then show the correct work.
Find Marcus's Mistake
- 1Set up the ratio3 cups : 5 cups = ? : 20 cups
- 2Find the multiplier20 ÷ 5 = 4
- 3Apply to both parts3 + 4 = 7
- 4Answer7 cups : 20 cups
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer: -
3Answer in complete sentences.
A store sells 3 pens for $2. Another store sells 9 pens for $6. Are these equivalent ratios? If so, prove it using at least two different methods (simplifying, scaling, or cross-multiplying).
Write your answer