3.1 Arithmetic, Order of Operations & Estimation
Key Takeaways
- Translate words into an expression before calculating, and label every quantity with its unit.
- Apply brackets, exponents, multiplication or division, then addition or subtraction in the correct order.
- Estimate the expected size and sign of an answer before exact arithmetic.
- Use inverse operations and substitution to check results instead of merely repeating the same calculation.
Arithmetic, order of operations and estimation
Represent the problem first
Many arithmetic errors begin before any numbers are combined. Read for the target, list what is known, and convert the relationship into an expression. Attach units.
Example: “A store has 18 boxes with 24 items in each box and distributes 75 items.” The remaining number is:
18 × 24 − 75
The multiplication must occur before subtraction because the stock is eighteen groups of twenty-four. Calculating 18 × (24 − 75) models a different story.
Useful translations include:
| Language | Operation or relationship |
|---|---|
| total, combined, increased by | addition |
| difference, fewer, remaining | subtraction |
| groups of, product, each | multiplication |
| shared equally, per group | division |
| of | often multiplication |
| per | often division or a rate |
Do not choose an operation from one keyword alone. “Eight fewer than twenty” means 20 − 8, while “subtract twenty from eight” means 8 − 20. Preserve the order expressed by the sentence.
Order of operations
Use this sequence:
- grouping symbols;
- powers or roots;
- multiplication and division from left to right;
- addition and subtraction from left to right.
Multiplication does not always precede division; they share a level. Thus 48 ÷ 6 × 2 = 8 × 2 = 16. Likewise, addition and subtraction proceed left to right: 20 − 7 + 3 = 13 + 3 = 16.
For 5 + 3(12 − 8)² ÷ 4:
- brackets: 12 − 8 = 4;
- exponent: 4² = 16;
- multiplication and division: 3 × 16 ÷ 4 = 12;
- addition: 5 + 12 = 17.
Write an intermediate line when the expression has more than one layer. Mental compression is fast only when it stays accurate.
Signed numbers
A negative sign can describe a direction, deficit, or value below a reference. For addition:
- same signs: add magnitudes and keep the sign;
- different signs: subtract magnitudes and keep the sign of the larger magnitude.
For multiplication or division, equal signs give a positive result and different signs give a negative result. Subtracting a negative is addition: 7 − (−4) = 11.
Use a number line when direction matters. If a value moves from −3 to 5, the change is 5 − (−3) = 8, not 2.
Zero, equality and inequality
Zero is neither positive nor negative, and division by zero is undefined. Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. For example, −2x > 8 becomes x < −4 after division by −2. Keep equality balanced by performing the same valid operation on both sides.
Absolute value measures distance from zero, so both 6 and −6 have absolute value 6. If a tolerance is written as |x − 20| ≤ 3, acceptable values lie no more than three units from 20: 17 ≤ x ≤ 23. Translate the notation back into a range before comparing options.
Mental strategies
Break numbers into convenient parts:
- 47 × 6 = (40 × 6) + (7 × 6) = 282;
- 198 + 357 = 200 + 357 − 2 = 555;
- 84 ÷ 7 can be recognised from 7 × 12 = 84;
- 25 × 48 = 100 × 12 = 1,200 by regrouping 25 × 4.
For multiplication by 5, multiply by 10 and halve. For division by 5, double and divide by 10. These are identities, not approximations.
Estimation as an error detector
Estimate before calculating. If 49 items cost $7.90 each, round to 50 × $8 = $400. An exact answer near $387 is plausible; $3,871 is a place-value error.
Use bounds when choices are close. If each quantity lies between 19 and 21 and there are six, the total lies between 114 and 126. Any option outside that interval is impossible.
Estimation also checks division. A quotient should be about 10 when a number near 600 is divided by a number near 60. If the exact work produces 100, inspect the decimal or divisor.
Remainders and rounding
Interpret remainders in context. If 97 people travel in vehicles that each hold 12, 97 ÷ 12 = 8 remainder 1, but nine vehicles are required. If 97 metres of material is cut into 12-metre lengths, only eight complete lengths are produced with one metre left.
Round only at the requested point. Carrying extra digits through intermediate work reduces cumulative error. State whether the answer needs a whole item, a decimal, or a measurement precision.
Verification
Check with an independent method:
- addition ↔ subtraction;
- multiplication ↔ division;
- substitute an answer into the original relationship;
- compare with the estimate;
- inspect units and sign.
If 18 × 24 − 75 = 357, check 357 + 75 = 432 and 432 ÷ 18 = 24. Repeating the same multiplication in the same arrangement may repeat the same mistake; an inverse check is stronger.
Evaluate 48 ÷ 6 × 2 + 5.
Ninety-seven people must travel in vehicles that hold at most 12 people each. What is the minimum number of vehicles?