Math101 · Independent Ontario learning library
Math101learn.math101.caSubtracting Fractions
Fractions can be subtracted only after they describe pieces of the same size. For $\frac ab-\frac cd$, rewrite both with a common nonzero denominator, then subtract numerators while preserving the common denominator.
Fraction subtraction compares measurements, changes, and probabilities. The common-unit reasoning transfers directly to algebraic rational expressions and unit conversion.
Intuition and core definition
Fractions can be subtracted only after they describe pieces of the same size. For $\frac ab-\frac cd$, rewrite both with a common nonzero denominator, then subtract numerators while preserving the common denominator. The result measures the signed difference between the two quantities.
Notation, language, and conditions
A least common denominator (LCD) is the LCM of the denominators. A negative sign may be attached to the numerator, denominator, or whole fraction. Parentheses matter when subtracting a negative fraction: $a-(-b)=a+b$. Mixed numbers may be converted to improper fractions or subtracted by regrouping.
Why this idea matters
Fraction subtraction compares quantities only after their parts are expressed in a common-sized unit, which is why a common denominator is structural rather than cosmetic.
A dependable method
- Check that all denominators are nonzero and convert mixed numbers if useful.
- Find the LCD of the denominators.
- Create equivalent fractions using the required scale factors.
- Subtract the numerators in the stated order and keep the LCD.
- Simplify, interpret the sign, and estimate with benchmark fractions.
Worked example
Representations and interpretation
On a number line, subtraction $a-b$ is the directed change from $b$ to $a$. Fraction strips must use a common partition; twelfths let both sixths and quarters align exactly before pieces are removed.
Reasoning about variations
If the second fraction is larger, the difference is negative: $1/3-5/6=2/6-5/6=-1/2$. The denominator still records piece size; subtracting denominators would compare incompatible units and produce a false partition.
Common mistakes
How to check your work
- Add the subtracted fraction to the difference and recover the starting fraction.
- Estimate using $0,1/2,1,$ and whole-number parts.
- Convert to decimals when they terminate and compare approximately.
Practice
- Compute $\frac78-\frac5{12}$.
- Compute $\frac25-\frac7{10}$.
- Compute $3\frac14-1\frac56$.
Answers and brief solutions
Show answers
- $\frac{11}{24}$ $21/24-10/24=11/24$.
- $-\frac3{10}$ $4/10-7/10=-3/10$.
- $1\frac5{12}$ $13/4-11/6=39/12-22/12=17/12=1\frac5{12}$.
Synthesis and transfer
Subtracting two elapsed times written as fractions of an hour first requires matching units; a timeline then checks the sign and approximate magnitude of the difference.
Suppose one interval lasts $5/6$ hour and another $3/8$ hour. Converting both to forty-eighths gives $40/48-18/48=22/48=11/24$ hour. The common denominator names equal-sized time pieces, making subtraction meaningful; adding the difference back to $3/8$ recovers $5/6$. A timeline shows that the result is positive and a little under one half, matching an estimate from approximately $0.83-0.38$. If the order is reversed, the magnitude is unchanged but the sign changes, so the wording “how much longer” must determine which quantity is the minuend. Units and direction are part of the answer.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Compute $\frac78-\frac5{12}$.
- $21/24-10/24=11/24$.
End of lesson
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