Order Rationals

Grade 6 · ratios · 88 practice problems · read aloud

🔊 Listen to this explanation

Ordering Rational Numbers (Ratios)

📘 What is Ordering Rationals?

Ordering rational numbers means arranging them from least to greatest or greatest to least. Rational numbers include fractions, decimals, and integers. This skill is super useful for comparing measurements, prices, and data in real life!

🛠️ How to Solve: Step-by-Step

  1. Convert to a Common Form: Change all numbers to decimals OR find a common denominator if using fractions.
  2. Compare Place Values: Look at the digits in each place value (ones, tenths, hundredths).
  3. Order Them: Arrange the numbers based on your comparison.

🔢 Visual Examples

Example 1: Order 0.75, 2/3, 0.7

  1. Convert: 2/3 ≈ 0.666..., 0.75, 0.7
  2. Compare: 0.666..., 0.7, 0.75
  3. Order (Least to Greatest): 2/3, 0.7, 0.75

Example 2: Order 1/2, 0.45, 3/5

  1. Convert (to fractions with denominator 20): 1/2 = 10/20, 0.45 = 9/20, 3/5 = 12/20
  2. Compare Numerators: 9, 10, 12
  3. Order (Greatest to Least): 3/5, 1/2, 0.45

⚠️ Common Mistakes

  • Assuming a larger denominator means a larger number. Remember: With the same numerator, a larger denominator makes a smaller fraction.
  • Not comparing decimals carefully. 0.5 is greater than 0.49, even though 49 looks bigger than 5. Compare digit by digit from the left!

💡 Tips & Tricks

  • Benchmark Method: Compare numbers to easy benchmarks like 0, 1/2, and 1.
  • Number Line: Draw a quick number line to visualize where each number falls.
  • Cross-Multiplication: To compare two fractions, cross-multiply. The larger product indicates the larger fraction.

🎯 Practice Suggestions

  • Use flashcards with a mix of fractions and decimals.
  • Play online math games that involve ordering numbers.
  • Practice with real-world examples, like ordering the lengths of different objects on a ruler.
  • Challenge yourself by ordering three or more numbers at a time!

Practice problems

6 of the 88, worked through step by step — try them before opening the answer.

1 (-3/4) + (2/3) = ?

Hint: When adding fractions with different denominators, first find a common denominator. Remember to handle negative signs carefully.

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 add the fractions. (-9/12) + (8/12) = (-9 + 8)/12 = -1/12.
  5. Conclusion. The sum is -1/12.

Let's add the fractions step by step.

2 (-3/4) + (5/6) = ?

Hint: When adding fractions with different denominators, first find a common denominator. Remember that negative fractions follow the same rules as positive fractions.

Show the answer

Answer: 1/12

  1. Identify the denominators. We have denominators 4 and 6.
  2. Find the least common denominator (LCD). Multiples of 4: 4, 8, 12, 16, ... Multiples of 6: 6, 12, 18, ... The smallest common multiple is 12. So the LCD 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 5/6: Multiply numerator and denominator by 2 → (5 × 2)/(6 × 2) = 10/12.
  4. Now add the fractions with the same denominator. (-9/12) + (10/12) = (-9 + 10)/12 = 1/12.
  5. Check if the result can be simplified. 1/12 is already in simplest form. Final answer: 1/12

Let's add the fractions step by step.

3 (-3/4) + (1/2) = ?

Hint: To add fractions with different denominators, first find a common denominator. Remember that negative numbers are less than zero.

Show the answer

Answer: -1/4

  1. Identify the problem. We are adding a negative fraction and a positive fraction: -3/4 + 1/2
  2. Find a common denominator. The denominators are 4 and 2. The least common denominator is 4.
  3. Rewrite the fractions with the common denominator. - The first fraction is already in terms of 4: -3/4 - The second fraction, 1/2, needs to be converted to have denominator 4. Multiply numerator and denominator by 2: (1 * 2) / (2 * 2) = 2/4
  4. Rewrite the problem with the common denominator. Now the problem is: (-3/4) + (2/4)
  5. Combine the numerators. Since the denominators are the same, we add the numerators and keep the denominator. (-3) + (2) = -1 So the result is: -1/4
  6. Final Answer. -1/4 Therefore, (-3/4) + (1/2) = -1/4

Let's solve: (-3/4) + (1/2) = ?

4 (-3.5 + 2.25) × 4 = ?

Hint: First combine the numbers inside the parentheses, then multiply the result by the number outside.

Show the answer

Answer: -5

  1. First, handle the addition inside the parentheses: -3.5 + 2.25 Think of this as 2.25 - 3.5. 2.25 - 3.5 = -1.25 So, (-3.5 + 2.25) = -1.25.
  2. Now multiply the result by 4: (-1.25) × 4 A negative times a positive gives a negative. 1.25 × 4 = 5.0 So, -1.25 × 4 = -5. Final Answer: -5

Let's solve step-by-step.

5 (-3.5) + 2.75 - (-1.25) = ?

Hint: Remember that subtracting a negative number is the same as adding its positive value. Work step by step from left to right.

Show the answer

Answer: 0.5

  1. Write the original problem clearly. (-3.5) + 2.75 - (-1.25)
  2. Interpret the subtraction of a negative number. Subtracting a negative is the same as adding a positive: - (-1.25) = + 1.25 So the expression becomes: (-3.5) + 2.75 + 1.25
  3. Add the positive numbers first for simplicity. 2.75 + 1.25 = 4.00
  4. Now add the negative number to the result. (-3.5) + 4.00 = 0.5
  5. Final answer: 0.5

Let's solve step-by-step:

6 (-2.5) + 3.75 - (-1.25) = ?

Hint: Remember that subtracting a negative number is the same as adding its positive value. Consider the direction each number moves you on a number line.

Show the answer

Answer: 2.5

  1. Write the original problem (-2.5) + 3.75 - (-1.25)
  2. Interpret the subtraction of a negative number Subtracting a negative number is the same as adding its positive: - (-1.25) = + 1.25 So the expression becomes: (-2.5) + 3.75 + 1.25
  3. Add the positive numbers first 3.75 + 1.25 = 5.00
  4. Now add the negative number to the result 5.00 + (-2.5) = 5.00 - 2.50 = 2.50
  5. Final answer 2.5 Therefore, (-2.5) + 3.75 - (-1.25) = 2.5

Let's solve step-by-step:

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