Fractions with Denominators of 10 & 100 - Fourth Grade Mathematics
In our base-ten number system, every whole-number place value column scales by powers of ten: ones, tens, hundreds, and thousands. It should come as no surprise that when mathematicians partition whole units into smaller pieces, the most natural, powerful denominators to choose are powers of ten as well: tenths and hundredths. Fractions with denominators of 10 and 100 serve as the vital bridge connecting common fractions directly to the decimal system. In this chapter, you will discover how tenths and hundredths relate to each other visually on ten-by-ten grids, learn how to convert tenths into hundredths with mathematical elegance, and build the foundation for decimal notation.
The Visual Anatomy of Tenths and Hundredths
A tenth represents one whole partitioned into 10 equal vertical strips, while a hundredth represents the exact same whole partitioned into 100 tiny square tiles.
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| TENTHS VERSUS HUNDREDTHS GRID COMPARISON |
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| MODEL A: ONE TENTH (1/10) MODEL B: TEN HUNDREDTHS (10/100) |
| (1 whole split into 10 strips) (1 whole split into 100 squares) |
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| +---+---+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+---+---+
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| +---+---+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+---+---+
| Shaded: 1 strip out of 10 = 1/10 Shaded: 10 squares out of 100 = 10/100
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| Notice: The shaded regions cover the EXACT SAME SPACE! |
| Therefore: 1/10 = 10/100 |
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The Ten-to-One Exchange Rate
Look closely at the two models above: each vertical strip of 1/10 contains exactly 10 individual hundredth-squares. 1 tenth = 10 hundredths (1/10 = 10/100) 2 tenths = 20 hundredths (2/10 = 20/100) 5 tenths = 50 hundredths (5/10 = 50/100) 7 tenths = 70 hundredths (7/10 = 70/100) This 10-to-1 ratio mirrors the exact same base-ten relationship found in whole numbers, where 1 ten equals 10 ones.
The Algebraic Conversion of Tenths to Hundredths
To convert any fraction with a denominator of 10 into an equivalent fraction with a denominator of 100, we apply the Identity Property of Multiplication by multiplying by 10/10.
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| CONVERTING TENTHS TO HUNDREDTHS |
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| Problem: Convert 4/10 to hundredths. |
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| 4 10 4 x 10 40 |
| --- x ---- = -------- = ----- |
| 10 10 10 x 10 100 |
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| Because 10/10 = 1 whole, multiplying 4/10 by 10/10 preserves its exact value! |
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| Quick Rule: Attach a zero to the numerator and a zero to the denominator: |
| 4/10 = 40/100 |
| 9/10 = 90/100 |
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Converting Hundredths to Tenths (Simplifying)
The conversion works in reverse by dividing both the numerator and denominator by 10/10: 60/100 = (60 / 10) / (100 / 10) = 6/10. 80/100 = (80 / 10) / (100 / 10) = 8/10. If a fraction of hundredths ends in a zero, it can always be simplified to tenths.
Chapter Practice Exercises
Exercise 1: Convert the following tenths into equivalent hundredths: 3/10, 6/10, and 8/10. Show the multiplication by 10/10 for each.
Exercise 2: Convert the following hundredths into equivalent tenths: 50/100, 70/100, and 20/100.
Exercise 3: Draw or describe a 10x10 hundredths grid showing the equivalence between 4/10 and 40/100.
Exercise 4: Explain why 7/10 is not equal to 7/100. Compare the sizes of the pieces using a ten-by-ten grid.
Exercise 5: A student wants to convert 2/10 to hundredths and writes 2/100, claiming they changed the denominator to 100. Explain why the student's fraction is ten times too small, and state the correct equivalent fraction.
Solutions and Step-by-Step Explanations
Solution 1: Multiplying by 10/10: 3/10 = (3 x 10) / (10 x 10) = 30/100. 6/10 = (6 x 10) / (10 x 10) = 60/100. 8/10 = (8 x 10) / (10 x 10) = 80/100.
Solution 2: Dividing numerator and denominator by 10: 50/100 = (50 / 10) / (100 / 10) = 5/10. 70/100 = (70 / 10) / (100 / 10) = 7/10. 20/100 = (20 / 10) / (100 / 10) = 2/10.
Solution 3: On a 10x10 grid with 100 equal squares, each full vertical column contains 10 squares and represents 1/10 of the whole grid. Shading 4 complete vertical columns represents 4/10. Counting the individual shaded squares inside those 4 columns: 4 columns x 10 squares per column = 40 squares shaded out of 100, which represents 40/100. Since the shaded region is identical, 4/10 = 40/100.
Solution 4: In 7/10, the whole is split into 10 large strips, and 7 full strips are shaded, which covers 70 out of 100 squares (more than half the grid!). In 7/100, the whole is split into 100 tiny squares, and only 7 individual tiny squares are shaded (less than one single strip!). Therefore, 7/10 is ten times larger than 7/100.
Solution 5: The student changed the denominator from 10 to 100 without changing the numerator. By multiplying only the denominator by 10, the student divided the value of the fraction by 10, making it ten times smaller! To maintain mathematical equivalence, the Identity Property requires multiplying the numerator by 10 as well (multiplying by 10/10): (2 x 10) / (10 x 10) = 20/100. Two tenths is equivalent to 20/100, not 2/100.