Adding Tenths & Hundredths Fractions - Fourth Grade Mathematics
When you look into your wallet and find three dimes and twenty-five pennies, you do not simply add three plus twenty-five to get twenty-eight dimes or twenty-eight pennies. Because dimes and pennies represent different denominations, you first convert the three dimes into thirty pennies, and then add thirty pennies plus twenty-five pennies to find a total of fifty-five cents. In fourth-grade mathematics, adding fractions with denominators of 10 and 100 requires the exact same logical thinking! Tenths and hundredths are unlike fractions that cannot be combined directly until they speak the same mathematical language. In this chapter, you will master the two-step strategy of converting tenths to hundredths to compute sums of tenths and hundredths with absolute confidence.
The Core Strategy: Convert Tenths to Hundredths First
To add a fraction with a denominator of 10 to a fraction with a denominator of 100, you must first rewrite the tenths fraction as an equivalent fraction with a denominator of 100.
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| ADDING TENTHS AND HUNDREDTHS: 3/10 + 45/100 |
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| Problem: 3/10 + 45/100 |
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| Step 1: CONVERT the tenths fraction to hundredths: |
| 3/10 = (3 x 10) / (10 x 10) = 30/100 |
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| Step 2: REWRITE the addition problem with like denominators: |
| 30/100 + 45/100 |
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| Step 3: ADD the numerators over the common denominator 100: |
| (30 + 45) / 100 = 75/100 |
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| Step 4: SIMPLIFY if applicable: |
| 75/100 = (75 / 25) / (100 / 25) = 3/4 |
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The Common Mistake: Adding Across Without Converting
The most dangerous error in fourth-grade fraction addition is adding numerators and denominators without converting: 3/10 + 45/100 = 48/110 (CATASTROPHIC ERROR!). Why is this mathematically meaningless? First, you cannot add denominators in fractions. Second, adding 3 tenths to 45 hundredths cannot produce "one-hundred-tenths." Tenths and hundredths are different-sized pieces. Converting 3/10 into 30/100 gives you 30 hundredths, allowing you to add 30 + 45 = 75 hundredths (75/100).
Visualizing Tenths and Hundredths Addition on a 100-Grid
An area model on a 10-by-10 grid provides clear visual proof of this addition process.
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| 10x10 GRID VISUALIZATION OF 4/10 + 27/100 |
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| Step 1: Shade 4 tenths (4 full vertical columns = 40 squares). |
| Step 2: Shade 27 hundredths (2 full vertical columns + 7 extra squares). |
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| Count total squares shaded: |
| 40 squares + 27 squares = 67 squares out of 100 total squares. |
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| Total Shaded Fraction: 67/100 |
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Sums Greater Than One: Mixed Numbers in Hundredths
When the sum of the hundredths exceeds 100, the result is an improper fraction that should be expressed as a mixed number.
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| SUMS GREATER THAN 1: 7/10 + 58/100 |
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| Step 1: Convert 7/10 to hundredths: 70/100 |
| Step 2: Add numerators: 70/100 + 58/100 = 128/100 |
| Step 3: Convert to a mixed number: |
| 128/100 = 100/100 + 28/100 = 1 and 28/100 |
| Step 4: Simplify: 1 and 28/100 = 1 and 7/25 |
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Chapter Practice Exercises
Exercise 1: Compute the sum: 4/10 + 38/100. Show the conversion step.
Exercise 2: Compute the sum: 6/10 + 52/100. Express your final answer as both an improper fraction and a mixed number.
Exercise 3: Compute the sum and simplify to lowest terms: 2/10 + 30/100.
Exercise 4: A science experiment requires 5/10 liter of water and 35/100 liter of saline solution. What is the total liquid volume needed for the experiment?
Exercise 5: A student calculates 5/10 + 23/100 and writes 28/100. Explain what step the student forgot and provide the correct sum.
Solutions and Step-by-Step Explanations
Solution 1: We convert 4/10 to hundredths by multiplying numerator and denominator by 10: 4/10 = 40/100. Now add: 40/100 + 38/100 = (40 + 38) / 100 = 78/100. Simplifying by dividing by 2 gives 39/50.
Solution 2: Convert 6/10 to hundredths: 6/10 = 60/100. Add: 60/100 + 52/100 = 112/100. As an improper fraction, the sum is 112/100. Converting to a mixed number: 112 / 100 = 1 with a remainder of 12 = 1 and 12/100 (which simplifies to 1 and 3/25).
Solution 3: Convert 2/10 to hundredths: 2/10 = 20/100. Add: 20/100 + 30/100 = 50/100. To simplify, divide numerator and denominator by their Greatest Common Factor of 50: (50 / 50) / (100 / 50) = 1/2. The simplified sum is 1/2.
Solution 4: To find the total volume, convert 5/10 liter to 50/100 liter: 50/100 + 35/100 = 85/100 liter. Simplifying by dividing by 5: (85 / 5) / (100 / 5) = 17/20 liter. The total liquid volume is 85/100 liter (or 17/20 liter).
Solution 5: The student forgot to convert 5/10 into 50/100 before adding. The student simply added the numerator 5 directly to the numerator 23 (5 + 23 = 28), treating 5 tenths as if it were 5 hundredths! In reality, 5 tenths equals 50 hundredths. The correct calculation is 50/100 + 23/100 = 73/100.