Did you know? The answer to −3/4 + 5/6 is a small positive fraction, even though the first number is negative. Rewrite both terms with denominator 12 and you obtain −9/12 + 10/12 = 1/12. A student who knows how to add fractions can still miss this question by losing the negative sign halfway through.
For Secondary 1 G3 Mathematics learners in Bukit Timah, mixed operations are often where several ordinary skills suddenly need to work together: equivalent fractions, negative signs, brackets, powers, multiplication and division. This guide is designed for quick understanding and reliable written working, not for memorising disconnected tricks.
Quick answer: solve what is grouped first; apply powers before ordinary multiplication and division; calculate multiplication and division from left to right; calculate addition and subtraction from left to right; and preserve the sign attached to each number. Convert fractions to a common denominator only when you are adding or subtracting them.
Jump to a fraction sign example, bracket calculation or quick answer key.
- Three rules that prevent most mixed-fraction mistakes
- Work out the sign before the arithmetic
- Worked example 1: fractions with opposite signs
- Worked example 2: subtracting a negative fraction
- Worked example 3: brackets change what we do first
- Worked example 4: a division inside brackets
- Worked example 5: mixed subtraction and multiplication
- Worked example 6: multiplication happens before subtraction
- Worked example 7: negative powers and brackets
- Fraction bars are grouping symbols
- Why the original answer should stay exact when possible
- Four mistakes that need different corrections
- The ten-question practice set
- Answers with a short reason
Three rules that prevent most mixed-fraction mistakes
- Add and subtract: first express the fractions using a common denominator. Then add or subtract the signed numerators without changing the denominator.
- Multiply: multiply the numerators and denominators, keeping track of the product’s sign. Cancel common factors when useful, not arbitrary terms separated by addition.
- Divide: multiply by the reciprocal of the nonzero divisor. For example, (−2/3) ÷ (4/5) = (−2/3) × (5/4) = −5/6.
The operation determines the fraction method. In 1/2 + 1/3, we need sixths; in (1/2) × (1/3), we can multiply directly to obtain 1/6.
Ask the learner to explain why a denominator changes. This is more valuable than learning a fixed set of button presses.
Work out the sign before the arithmetic
A negative plus a positive does not have an automatic sign. Compare their magnitudes. In −3/4 + 5/6, the positive 5/6 is larger in magnitude than 3/4, so the final result is positive.
Two negative numbers added together give a negative sum: −1/2 + (−1/4) = −3/4.
Multiplying two negative factors gives a positive result: (−2/3) × (−3/5) = 2/5.
Dividing a negative by a positive gives a negative result: (−3/4) ÷ (1/2) = −3/2.
If the final sign contradicts an estimate, inspect the first step at which a sign was copied or changed. For more sign-specific work, see negative numbers without sign errors.
Worked example 1: fractions with opposite signs
Evaluate −3/4 + 5/6.
First, find the lowest common denominator of 4 and 6, which is 12. Rewrite −3/4 as −9/12 and 5/6 as 10/12.
Then −9/12 + 10/12 = 1/12. The answer is positive because ten twelfths outweighs nine twelfths by one twelfth.
Check with decimals: −0.75 + approximately 0.8333 is approximately 0.0833, matching 1/12.
An incorrect answer of −19/12 may come from adding the magnitudes and keeping the first negative sign. The opposite signs require a signed sum, not merely adding numerator sizes.
Worked example 2: subtracting a negative fraction
Calculate 3/5 − (−1/4).
Subtracting a negative number is equivalent to adding its opposite. Therefore 3/5 − (−1/4) = 3/5 + 1/4.
Use denominator 20: 12/20 + 5/20 = 17/20.
Check the direction: subtracting a negative quarter should make the starting positive three fifths larger, so 17/20 is plausible.
Compare a changed question, 3/5 − 1/4. Without the second negative sign, the answer is 12/20 − 5/20 = 7/20. One sign changes the entire relationship.
Worked example 3: brackets change what we do first
Evaluate −(2/3)(9/4 − 3/2).
The bracket contains subtraction. Rewrite 3/2 as 6/4, giving 9/4 − 6/4 = 3/4.
Now multiply by negative two thirds: −(2/3) × (3/4) = −6/12 = −1/2.
Notice that the negative sign applies to the whole product. We did not need to insert an extra minus into each bracket term before computing it.
Check numerically: the bracket is 0.75, and negative two thirds of 0.75 is −0.5.
Worked example 4: a division inside brackets
Evaluate 1/2 − [(3/4) ÷ (−3/2)].
Do the bracketed division first. Dividing by −3/2 means multiplying by −2/3. Thus (3/4) × (−2/3) = −1/2.
Now 1/2 − (−1/2) = 1.
If the pupil reports zero, check whether they treated the bracket’s negative result as a positive fraction. If they report a negative value, ask them to explain the final subtraction of a negative half.
The operation in the bracket must finish before the outer subtraction takes place.
Worked example 5: mixed subtraction and multiplication
Evaluate [−5/6 − (−1/3)] × (−9/2).
Inside the bracket, subtracting negative one third means adding one third. With denominator six, we get −5/6 + 2/6 = −3/6 = −1/2.
Multiply by −9/2: (−1/2) × (−9/2) = 9/4.
The result is positive because the two factors being multiplied are negative. It is also greater than two: 9/4 = 2 1/4.
A sensible approximate check is (−0.5) × (−4.5) = 2.25. The fraction and decimal agree.
Worked example 6: multiplication happens before subtraction
Evaluate 2 − (3/4) × (8/3).
Multiply the fractions first. The factor 3 cancels against the 3 in the denominator; 8 divided by 4 leaves 2. Thus (3/4) × (8/3) = 2.
Now calculate 2 − 2 = 0.
It would be wrong to calculate 2 − 3/4 first and then multiply by 8/3. That would be the different expression (2 − 3/4) × (8/3), whose value is 10/3.
This is why the visual position of brackets is meaningful. They show which subexpression must be treated as one unit.
Worked example 7: negative powers and brackets
Compare −2² with (−2)².
In −2², the power applies to the 2 before the leading negative sign, giving −(2²) = −4.
In (−2)², the bracket makes the entire negative number the base: (−2)(−2) = 4.
Now evaluate −2² + (−2)² = −4 + 4 = 0.
A calculator is useful for checking only when the input preserves the intended brackets. The calculator and BODMAS guide explains why input structure matters.
Fraction bars are grouping symbols
The expression (1 + 1/2)/(3/4) has an entire numerator 1 + 1/2 and one denominator 3/4.
Compute the numerator first: 1 + 1/2 = 3/2. Then (3/2) ÷ (3/4) = (3/2) × (4/3) = 2.
Without clear grouping, typing 1 + 1/2 ÷ 3/4 may tell a calculator to divide only part of the expression. That is not the same mathematical task.
When copying a printed fraction, use clear brackets around multi-term numerators or denominators in linear-entry notation.
Why the original answer should stay exact when possible
The fraction 1/3 equals a repeating decimal 0.3333…, but writing 0.33 replaces it with a rounded approximation.
Consider 1/3 + 1/3 + 1/3. Exactly, the result is 1. If you round each third to 0.33 before adding, you get 0.99.
Unless the question instructs otherwise, use exact fractions through intermediate steps and round only the final answer if a decimal accuracy is requested.
An estimate is still useful. If a mixed expression contains several large positive terms and one tiny negative fraction, a final result with the opposite large sign deserves checking.
Four mistakes that need different corrections
Common-denominator error: writing 1/2 + 1/3 = 2/5. The pieces being added are not the same size.
Sign error: writing 3/5 − (−1/4) = 3/5 − 1/4. The second negative was not interpreted.
Grouping error: calculating an outside subtraction before finishing a bracketed division.
Cancellation error: crossing out equal-looking pieces from sums instead of cancelling genuine multiplicative factors.
These wrong answers can look similar on a marked worksheet. A tutor should ask for the first incorrect line so the correction targets the cause.
The ten-question practice set
Work without a calculator first when the fractions are manageable, then verify the results. Explain one sign decision in words.
- A. Evaluate −1/2 + 3/4.
- B. Evaluate −3/4 + 5/6.
- C. Evaluate 3/5 − (−1/4).
- D. Evaluate (−2/3) × (9/4).
- E. Evaluate (−5/6) ÷ (2/3).
- F. Evaluate −(2/3)(9/4 − 3/2).
- G. Evaluate 1/2 − [(3/4) ÷ (−3/2)].
- H. Evaluate 2 − (3/4) × (8/3).
- I. Evaluate −2² + (−2)².
- J. Evaluate [−5/6 − (−1/3)] × (−9/2).
Answers with a short reason
A. 1/4; −2/4 + 3/4 = 1/4.
B. 1/12; use denominator 12.
C. 17/20; subtracting a negative adds its opposite.
D. −3/2; the product of a negative and positive is negative.
E. −5/4; multiply by the reciprocal of the positive divisor.
F. −1/2; calculate the bracket first.
G. 1; the bracket gives −1/2, which is then subtracted.
H. 0; perform multiplication before subtraction.
I. 0; the bracket changes which quantity is squared.
J. 9/4; two negative factors multiply to a positive result.
An accurate answer is useful evidence, but so is a clear explanation of why it is positive, negative or zero. Both should be checked.
A focused fifteen-minute home lesson
For the first four minutes, compare three signed fractions on a number line. Then use four minutes for fraction addition and multiplication, asking which methods require a common denominator.
Use five minutes for two mixed operations with brackets and negative signs. Ask the child to mark the first operation before calculating.
Spend the last two minutes inspecting an incorrect solution and fixing the first invalid line. Revisit a similar but changed question a few days later, without the model visible.
The aim is accurate independent decisions, not merely a growing number of completed questions.
What a Bukit Timah Mathematics tutor can observe
A helpful tutor separates weak fraction equivalence, incorrect sign reading, bracket mistakes and calculation-order errors. These may need different repairs even though they all appear in one question.
The immutable eduKateSG Secondary 1 Mathematics small-group reference describes focused tutorials near Sixth Avenue MRT. The Bukit Timah G3 Mathematics tuition overview adds local programme context. Families should check current suitability and class details directly.
The official MOE G2 and G3 Mathematics syllabus provides the curriculum context for rational numbers, signed operations, estimation and algebraic notation. Schools may sequence practice differently.
Frequently asked questions
### Must I use a common denominator when multiplying fractions?
No. Multiply the numerators and denominators directly, simplifying shared factors where valid. Common denominators are mainly needed for fraction addition and subtraction.
### Why does subtracting a negative become addition?
Subtracting −a is equivalent to adding a. The difference can be understood on a number line or by the inverse relationship between addition and subtraction.
### Why shouldn’t I round the fractions too early?
Rounded intermediates can introduce avoidable error. Keep fractions exact where practical and follow the requested final accuracy.
### Should a student use a calculator for every mixed operation?
A calculator can check arithmetic when permitted, but interpreting signs, brackets and fraction operations remains the student’s responsibility.
### What if the answer is a negative fraction?
Check it against the original expression. Negative results can be completely valid; the final sign should follow the operations and the relative magnitudes.
Continue learning through the series
- Rational and irrational numbers, fractions and number lines
- Calculator skills, BODMAS and bracket checks
- Negative numbers without sign errors
- Fractional coefficients in algebraic expressions
- Parallel-line angles and transversals
The useful question after every line is this: “Does my new expression still mean exactly the same thing as the original?” Once the answer stays yes, mixed fractions become far less intimidating.