Rational Properties

Grade 7 · ratios · 73 practice problems · read aloud

🔊 Listen to this explanation

What Are Ratios? 🤔

A ratio is a comparison between two or more quantities. Ratios help us understand how much of one thing there is compared to another. We use ratios in recipes, maps, sports statistics, and many real-life situations!

Example: If a class has 12 girls and 8 boys, the ratio of girls to boys is 12:8.

How to Work with Ratios 📝

  1. Write the ratio using colon notation (:) or as a fraction
  2. Simplify by dividing both numbers by their greatest common factor
  3. Check your answer - simplified ratios should have whole numbers

Worked Examples

Example 1: Simplifying Ratios

Problem: Simplify 16:24

Step 1: Find the greatest common factor (GCF) of 16 and 24 → 8

Step 2: Divide both numbers by 8 → 16÷8 = 2, 24÷8 = 3

Answer: 2:3

Example 2: Equivalent Ratios

Problem: Find two equivalent ratios to 3:5

Step 1: Multiply both terms by 2 → 3×2:5×2 = 6:10

Step 2: Multiply both terms by 3 → 3×3:5×3 = 9:15

Answer: 6:10 and 9:15 are equivalent to 3:5

⚠️ Common Mistakes to Avoid

  • Forgetting to simplify - Always reduce ratios to simplest form
  • Mixing up order - The ratio 2:3 is different from 3:2
  • Using decimals - Simplified ratios should use whole numbers
  • Incorrect GCF - Double-check your greatest common factor

🌟 Tips & Tricks

  • GCF shortcut: If both numbers are even, you can always divide by 2
  • Memory aid: "What's the biggest number that divides evenly into both?"
  • Check your work: Multiply your simplified ratio - it should match the original when scaled up

Practice Makes Perfect! 🎯

Try these practice ideas:

  • Simplify these ratios: 18:24, 15:35, 28:42
  • Find three equivalent ratios for 4:7
  • Look for ratios in your daily life (recipe measurements, sports statistics)
  • Create ratio word problems about your favorite hobbies

Practice problems

6 of the 73, worked through step by step — try them before opening the answer.

1 -3/4 + 2/3 = ?

Hint: When adding fractions with different denominators, find a common denominator first. Remember to apply the negative sign correctly to the numerator.

Show the answer

Answer: -1/12

  1. Identify the denominators. The denominators are 4 and 3.
  2. Find a common denominator. The least common multiple of 4 and 3 is 12.
  3. Rewrite each fraction with denominator 12. For -3/4: Multiply numerator and denominator by 3 → (-3 × 3)/(4 × 3) = -9/12. For 2/3: Multiply numerator and denominator by 4 → (2 × 4)/(3 × 4) = 8/12.
  4. Now the problem is: -9/12 + 8/12
  5. Add the numerators over the common denominator. (-9 + 8)/12 = (-1)/12
  6. Simplify if possible. -1/12 is already in simplest form. Final answer: -1/12

Let's add -3/4 + 2/3 step-by-step.

2 (-15) × [(-20) + 35] = ?

Hint: Think about the distributive property of multiplication over addition. Apply the multiplication to each term inside the brackets separately, then combine the results.

Show the answer

Answer: -225

  1. Use the distributive property: a × (b + c) = a × b + a × c Here, a = -15, b = -20, c = 35
  2. Compute (-15) × (-20) = 300 (negative × negative = positive)
  3. Compute (-15) × 35 = -525 (negative × positive = negative)
  4. Add the results: 300 + (-525) = -225

The answer is -225.

3 (-25) × [(-30) + 45] = ?

Hint: Think about the distributive property of multiplication over addition. You can multiply the number outside the brackets by each term inside the brackets separately, then combine the results. Pay close attention to the negative signs.

Show the answer

Answer: -375

  1. Use the distributive property: a × (b + c) = a × b + a × c. Here, a = -25, b = -30, c = 45.
  2. Compute (-25) × (-30) = 750 (negative × negative = positive).
  3. Compute (-25) × 45 = -1125 (negative × positive = negative).
  4. Add the results: 750 + (-1125) = -375.

The answer is -375.

4 (-17) × [(-33) + 51] = ?

Hint: Think about the distributive property: multiply the number outside the brackets by each term inside the brackets separately, then combine the results. Pay close attention to the signs when multiplying.

Show the answer

Answer: -306

  1. Use the distributive property: a × (b + c) = a × b + a × c Here, a = -17, b = -33, c = 51
  2. Compute (-17) × (-33) = 561 (negative × negative = positive)
  3. Compute (-17) × 51 = -867 (negative × positive = negative)
  4. Add the results: 561 + (-867) = 561 - 867 = -306

The answer is -306.

5 (-24) × [(-36) + 48] = ?

Hint: Think about the distributive property: multiply the number outside the brackets by each term inside separately, then combine the results. Pay close attention to the signs when multiplying.

Show the answer

Answer: -288

  1. Use the distributive property: a × (b + c) = a × b + a × c Here, a = -24, b = -36, c = 48
  2. Compute (-24) × (-36) = 864 (negative × negative = positive)
  3. Compute (-24) × 48 = -1152 (negative × positive = negative)
  4. Add the results: 864 + (-1152) = 864 - 1152 = -288

The answer is -288.

6 (-14) × [(-28) + 42] = ?

Hint: Think about the distributive property of multiplication over addition. Apply the multiplication to each term inside the brackets separately, then combine the results. Pay close attention to the signs when multiplying negatives.

Show the answer

Answer: -196

  1. Use the distributive property: a × (b + c) = a × b + a × c Here, a = -14, b = -28, c = 42
  2. Compute (-14) × (-28) = 392 (negative × negative = positive)
  3. Compute (-14) × 42 = -588 (negative × positive = negative)
  4. Add the results: 392 + (-588) = 392 - 588 = -196

The answer is -196.

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