A long numerical expression becomes manageable when you keep its structure visible. The aim is to replace one complete part at a time with an equal value, while leaving every unfinished operation intact. A misplaced bracket can change the question even when the arithmetic that follows is accurate.
This lesson combines signed integers, fractions and terminating decimals. The original examples use explicit multiplication and division signs. Before calculating, read the entire expression, identify any complete numerator and denominator, and check what the final operation will be.
Read the grouping before choosing an operation
Brackets tell you which values belong together. A fraction bar also groups: (a + b)/(c − d) means divide the whole numerator by the whole denominator. Written with a division sign, the same structure is (a + b) ÷ (c − d). Omitting either pair of brackets can create a different calculation.
Within a group, multiplication and division have equal priority and proceed from left to right. Addition and subtraction likewise have equal priority and proceed from left to right after the products and quotients. Do not interpret a remembered list of letters as an instruction to do every multiplication before every division.
A negative sign can belong to a number, while a subtraction sign describes an operation between quantities. In 7 − (−1), you subtract the number negative one. This is 7 + 1. By contrast, 7 + (−1) is 6. Keep the brackets until you have deliberately applied the sign rule.
Worked case 1: a signed numerator over a grouped denominator
Evaluate [ (−18) ÷ 3 + 4 ] / [ 7 − (−1) ]. Here the final operation is division between two groups. First evaluate the numerator: (−18) ÷ 3 + 4 = −6 + 4 = −2. The denominator is 7 + 1 = 8, which is nonzero.
The complete quotient is therefore −2/8 = −1/4. A decimal check gives −0.25. Since the numerator is negative and the denominator positive, the result must be negative. Its magnitude is smaller than one because the numerator’s magnitude, 2, is smaller than 8.
Text route through the diagram: calculate the numerator branch to get −2; calculate the denominator branch to get 8; divide those branch results to get −1/4. Neither branch can be replaced by only its first number. In particular, dividing −18 by the final denominator before completing the numerator would change the grouping.
Choose an exact number form deliberately
Terminating decimals can be converted exactly to fractions using place value: 1.2 = 12/10 = 6/5 and 0.375 = 375/1000 = 3/8. Fractions are useful when a mixture contains thirds or other non-terminating decimal values. Replacing 2/3 by 0.67 introduces an approximation before you know whether it is acceptable.
Decimals may be quicker when all divisions terminate cleanly. The best form is the one that makes the operations and checks clearest. Conversion changes the representation, not the value. Reduce fractions when convenient, but cancel only common factors in a product, never unrelated terms separated by addition.
Worked case 2: subtract a quotient containing decimals
Evaluate 3/4 − [ (1.2 − 7/5) ÷ (−2/3) ]. Convert 1.2 to 6/5. The inner subtraction is 6/5 − 7/5 = −1/5. Dividing by −2/3 means multiplying by its reciprocal, −3/2, so the bracket becomes (−1/5) × (−3/2) = 3/10.
Now complete the outer subtraction: 3/4 − 3/10 = 15/20 − 6/20 = 9/20 = 0.45. The quotient is positive because both dividend and divisor are negative. That positive quotient is then subtracted from 3/4; the minus sign outside the bracket still matters.
For a different check, 1.2 − 1.4 = −0.2 exactly. Multiplying this by −1.5, the exact reciprocal of −2/3, gives 0.3. Finally 0.75 − 0.3 = 0.45. This check uses terminating decimals throughout and agrees with the exact fraction, without rounding two-thirds.
Worked case 3: locate the first invalid transformation
A learner writes 30 ÷ 5 × 2 − 7 = 30 ÷ 10 − 7 = −4. The last arithmetic is correct for the altered expression, but the first equality is false. Multiplying 5 by 2 first has inserted grouping that the original question did not contain.
The valid chain is 30 ÷ 5 × 2 − 7 = 6 × 2 − 7 = 12 − 7 = 5. A useful cross-check rewrites division as multiplication by a reciprocal: 30 × (1/5) × 2 − 7 gives the same result. The expression 30 ÷ (5 × 2) − 7 really would be −4, but its brackets make it a different question.
When diagnosing an error, name the first changed meaning rather than merely crossing out the final number. “An unsupported pair of brackets was inserted around 5 × 2” tells the learner what to repair and why.
Three solved practice tasks
Practice 1: keep both sign decisions
Evaluate [ 5 − (−3) ] ÷ [ 2 − 6 ]. The first bracket is 8 and the second is −4, so the answer is −2. The denominator is nonzero. Multiplying the result back by −4 returns 8, which checks the final division independently.
Practice 2: verify the fraction and decimal
Evaluate (0.875 − 1/4) ÷ (5/2). In fractions, (7/8 − 2/8) × (2/5) = (5/8) × (2/5) = 1/4. In decimals, 0.625 ÷ 2.5 = 0.25. These agree exactly. Dividing by a number greater than one reduces the positive input, so 2.5 or 1.5625 would fail a size check.
Practice 3: check whether the question has a value
Evaluate [ 3/5 + 0.4 ] / [ 1.5 − 3/2 ], if defined. The numerator is 1, but the denominator is 3/2 − 3/2 = 0. The expression is undefined. It is neither zero nor one. No number multiplied by zero can return the numerator 1, so the requested division cannot be performed.
A final check that tests the calculation
Read your working from top to bottom and ask whether every equality preserves the original expression. Then check sign, rough size and denominator separately. A calculator can verify a correctly entered expression, but it cannot tell you that you accidentally omitted the brackets around a complete denominator. Enter the grouped structure first, then compare the answer with your exact working.
Frequently asked questions
Does multiplication always come before division?
No. They have equal priority. Work from left to right unless brackets or a fraction bar establish a different grouping.
What does the fraction bar group?
It groups the complete numerator and complete denominator. Evaluate each as a whole before dividing their values.
Why does subtracting a negative increase a number?
Subtracting a number means adding its opposite. The opposite of a negative number is positive, so 7 − (−1) becomes 7 + 1.
Must every decimal be converted to a fraction?
No. Choose a convenient exact form. Fractions are particularly useful when decimal conversion would require rounding a recurring value.
When do I invert a fraction?
When dividing by a nonzero fraction, multiply by its reciprocal. Do not invert the dividend or every fraction in the expression.
Can a zero numerator be allowed?
Yes, provided the denominator is nonzero: zero divided by a nonzero number is zero. A zero denominator is never allowed here.
Is a plausible decimal enough to verify my work?
No. Check the grouping and each transformation as well. A different expression can produce a plausible answer even after accurate arithmetic.
Subject references
The original questions, diagrams and solutions above use standard mathematical definitions and methods. These source-authored references support the underlying facts.
WhatsApp Us Now