Fractions · Lesson 7 of 9
Subtracting Fractions
“Find remaining amounts and differences by expressing fractions in the same unit before subtracting.”
• Explain subtraction as removing equal-sized pieces or moving back along a number line. • Subtract fractions with matching denominators. • Use equivalent fractions and Brahmagupta’s method for unlike denominators. • Interpret the direction of subtraction correctly. • Solve distance and time problems and simplify the resulting difference.
Remove some of the pieces you have
Suppose a strip has six of its seven equal parts shaded. If four of those seventh-pieces are removed, two seventh-pieces remain. The calculation is 6/7 − 4/7 = 2/7. Subtraction changes how many pieces we have, while the size of a piece stays one seventh. This is the same distinction between count and unit that made addition work.
Problem
Find 6/7 − 4/7.
- 1.Start with six pieces of size 1/7.
- 2.Remove four of those same-sized pieces.
- 3.The number left is 6 − 4 = 2, so the result is 2/7.
A number line gives another interpretation. Begin at the position 6/7, then move left by four intervals of length 1/7. You land at 2/7. A remaining quantity and a backward distance describe the same subtraction. In either picture, the two amounts must be measured against the same whole unit.
With a common denominator, subtract the numerators because they count the same-sized units. The denominator remains fixed. The relationship below is written for a at least as large as c, so the result is zero or a positive amount. This lesson’s problems use that situation.
Problem
Find 5/8 − 3/8 and give the answer in lowest terms.
- 1.Both amounts use eighth-pieces.
- 2.Subtract the counts: 5 − 3 = 2, so the result is 2/8.
- 3.Two eighths group into one quarter. Dividing both numbers by 2 gives 1/4.
Problem
Find 10/27 − 1/27.
- 1.The two amounts use the same unit, 1/27.
- 2.Removing one piece from ten leaves nine, giving 9/27.
- 3.Both 9 and 27 have factor 9. Divide by 9 to obtain 1/3.
6/7 − 4/7 is not 2/0. Removing seventh-pieces cannot erase the denominator. The difference counts the pieces remaining, which are still sevenths.
Make unequal units match before removing them
To subtract two thirds from three quarters, quarter-pieces and third-pieces are not yet comparable counts. Change both descriptions to twelfths. Three quarters becomes nine twelfths, while two thirds becomes eight twelfths. We can then remove eight twelfth-pieces from nine and see exactly what remains.
Problem
Find 3/4 − 2/3.
- 1.Choose common denominator 12.
- 2.Multiply both numbers in 3/4 by 3 to obtain 9/12.
- 3.Multiply both numbers in 2/3 by 4 to obtain 8/12.
- 4.Subtract: 9/12 − 8/12 = 1/12. One twelfth is already in lowest terms.
The method is the subtraction form of Brahmagupta’s common-denominator procedure. Find an appropriate common multiple, preserve each fraction while changing its unit, then subtract the numerator counts. Simplify the difference when possible. As with addition, a lowest common multiple is useful, but any common multiple gives the same final amount.
• Express the two fractions with a common denominator. • Subtract the second numerator from the first and retain the denominator. • Reduce the result to lowest terms when possible. • Check whether the answer represents a remaining amount or the difference between two amounts.
Problem
Find 2/5 − 4/15.
- 1.Fifteen is already a multiple of both denominators.
- 2.Change 2/5 to 6/15 by multiplying numerator and denominator by 3.
- 3.Subtract 4/15 from 6/15 to get 2/15.
- 4.The numerator 2 and denominator 15 have no common factor greater than 1, so stop.
Problem
Find 5/6 − 4/9.
- 1.Use common denominator 18.
- 2.Write 5/6 = 15/18 and 4/9 = 8/18.
- 3.Subtract the counts: 15 − 8 = 7, giving 7/18.
- 4.Since 7 and 18 have no common factor greater than 1, the difference is in lowest terms.
Read the direction of subtraction
“Subtract 13/4 from 10/3” means begin with 10/3 and take away 13/4. The word “from” identifies the starting amount. Do not subtract merely in the order the fractions appear in the sentence. For a difference between two measured quantities, first compare them so you know which one is larger.
Problem
Subtract 13/4 from 10/3.
- 1.Translate the instruction as 10/3 − 13/4.
- 2.Use twelfths: 10/3 = 40/12 and 13/4 = 39/12.
- 3.Subtract to get 1/12. Both starting amounts exceeded three wholes, but their difference is small.
Subtraction can leave more than one whole. For example, subtracting 18/5 from 23/3 gives 115/15 − 54/15 = 61/15. Four groups of fifteen use sixty of the pieces, leaving one fifteenth, so the difference is 4 1/15. The method is unchanged when the numerator is greater than the denominator.
Distances and times give subtraction meaning
If a journey is partly completed, subtract the completed length from the full length to find what remains. If two people take different times for the same trip, subtract the shorter time from the longer time to measure their time difference. Keep the units consistent: kilometres must be compared with kilometres, and minutes with minutes.
Problem
Jaya’s school is 7/10 km from home. She rides an auto for 1/2 km and walks the rest. How far does she walk?
- 1.The distance walked is the total distance minus the auto distance: 7/10 − 1/2.
- 2.Rewrite the half-kilometre as 5/10 km.
- 3.Subtract: 7/10 − 5/10 = 2/10 km.
- 4.Simplify to 1/5 km. Check that 5/10 km ridden plus 2/10 km walked makes the complete 7/10 km journey.
Problem
Jeevika takes 10/3 minutes for a round of the park. Namit takes 13/4 minutes. Who takes less time and by how much?
- 1.Express the times in twelfths of a minute: 10/3 = 40/12 and 13/4 = 39/12.
- 2.Namit’s time is smaller because 39 < 40.
- 3.Subtract the smaller time from the larger: 40/12 − 39/12 = 1/12 minute.
- 4.Namit takes 1/12 minute less than Jeevika. The answer names both the person and the size of the difference.
Quiz
What is 6/7 − 4/7?
What is 5/8 − 3/8 in lowest terms?
Which calculation correctly represents 3/4 − 2/3?
Subtract 13/4 from 10/3 means…
After riding 1/2 km of a 7/10 km journey, the remaining distance is…
Which is correct for 10/3 minutes and 13/4 minutes?
Practice Problems
- Use a shaded strip and a number line to explain 6/7 − 4/7. Describe what stays unchanged during removal.
- Subtract and simplify: 5/8 − 3/8; 7/9 − 5/9; 10/27 − 1/27.
- Use a common denominator to find 8/15 − 3/15; 2/5 − 4/15; 5/6 − 4/9; 2/3 − 1/2.
- Subtract 13/4 from 10/3; 18/5 from 23/3; 29/7 from 45/7. State the starting amount in each calculation.
- Solve Jaya’s school-travel problem with a labelled length diagram. Check the result by adding the two journey parts.
- Compare Jeevika’s and Namit’s times around the park. Explain why a smaller time means completing the round sooner.
- A student writes 3/4 − 2/3 = 1 because both top numbers and both bottom numbers differ by one. Explain why the units must first match.
- A ribbon is 1 1/4 metres long and 2/3 metre is used. Find the remaining length by first converting the mixed number to a fraction.
Key Takeaways
• Subtraction removes or compares counts of the same fractional unit. • With equal denominators, subtract numerators and keep the denominator. • With different denominators, use equivalent fractions before subtracting. • “Subtract a from b” means start with b and calculate b − a. • A practical answer should include its measurement unit and describe what the difference means.