C1.6; E1.6 - Order of operations

C1.6; E1.6 - Order of operations

Does 3+2×523+2\times5^2 have the same value as (3+2)×52(3+2)\times5^2? A small pair of brackets changes which calculation happens first. C1.6 and E1.6 use a common order for brackets, powers and the four operations, so integer, fraction and decimal calculations have one unambiguous value.

Brackets and powers

Brackets tell you to treat an expression as one quantity. Work inside the innermost brackets first. After brackets, evaluate powers before multiplication or division, then carry out addition or subtraction.

Order of operations

Work in this order: brackets, then powers, then multiplication and division, then addition and subtraction. Operations of equal priority are handled from left to right.

A fraction bar also groups its complete numerator and denominator. In 8+42\frac{8+4}{2}, calculate the top 8+48+4 before dividing. This differs from 8+428+\frac42. In nested brackets, 2[7−(3+1)]2[7-(3+1)] means find 3+13+1, subtract it from 7, then multiply by 2.

Worked example 1: a power without outer brackets

Question. Work out 3+2×523+2\times5^2.

Plan. Evaluate the square, then the product, then the sum.

3+2×52=3+2×25power first=3+50multiplication next=53\begin{aligned} 3+2\times5^2&=3+2\times25 &&\text{power first}\\ &=3+50 &&\text{multiplication next}\\ &=53 \end{aligned}

Answer. 53.

Check. Two groups of 25 make 50; adding the separate 3 gives 53. The 3 was not part of the multiplication.

The next expression changes only the brackets. Their position makes the addition part of the quantity being multiplied.

Worked example 2: brackets change the value

Question. Work out (3+2)×52(3+2)\times5^2.

Plan. Complete the bracket and the power, then multiply their values.

(3+2)×52=5×25bracket and power=125\begin{aligned} (3+2)\times5^2&=5\times25 &&\text{bracket and power}\\ &=125 \end{aligned}

Answer. 125.

Check. Five groups of 25 total 100+25=125100+25=125. This is different from 53 because the 3 is now included in the five groups.

A negative sign needs the same care: (−3)2=9(-3)^2=9 squares the complete negative number, whereas −32=−(32)=−9-3^2=-(3^2)=-9. Brackets show exactly what the power applies to. Write each simplified expression on a new line, preserving any brackets still needed.

Watch out

Working from left to right across every operation would give 3+2×25=5×253+2\times25=5\times25, which is wrong. Left-to-right applies within one priority level, after higher-priority operations are complete.

Start with the bracket, then square its value.

Equal-priority operations

Multiplication and division share one priority level. Addition and subtraction share another. Their order in a memory aid does not give multiplication priority over division, or addition priority over subtraction.

For example, 12÷3×212\div3\times2 means (12÷3)×2=8(12\div3)\times2=8. It is different from 12÷(3×2)=212\div(3\times2)=2, where brackets explicitly group the divisor. Similarly, 10−4+1=(10−4)+1=710-4+1=(10-4)+1=7.

Worked example 3: division followed by multiplication

Question. Work out 24÷6×2+5−324\div6\times2+5-3.

Plan. Complete division and multiplication from left to right, then addition and subtraction from left to right.

24÷6×2+5−3=4×2+5−3=8+5−3=13−3=10\begin{aligned} 24\div6\times2+5-3&=4\times2+5-3\\ &=8+5-3\\ &=13-3\\ &=10 \end{aligned}

Answer. 10.

Check. The multiplicative part is two copies of 24÷6=424\div6=4, so it is 8. The remaining +5−3+5-3 adds 2 overall, giving 10.

The next calculation puts subtraction first in the written expression. A later division still has higher priority, but once it is complete, the remaining subtraction and addition go left to right.

Worked example 4: subtraction followed by addition

Question. Work out 18−9+12÷3×218-9+12\div3\times2.

Plan. Evaluate the division-and-multiplication chain first; then process the remaining sum and difference in order.

18−9+12÷3×2=18−9+4×2=18−9+8=9+8=17\begin{aligned} 18-9+12\div3\times2&=18-9+4\times2\\ &=18-9+8\\ &=9+8\\ &=17 \end{aligned}

Answer. 17.

Check. 12÷3×2=812\div3\times2=8. The expression becomes 18+(−9)+818+(-9)+8, whose signed terms total 17. Grouping 9+89+8 as a quantity to subtract would change the expression.

For a longer chain, simplify one step at a time. A new line should still be equal to the previous line; do not use a running chain such as 24÷6=4×2=824\div6=4\times2=8, because 24÷624\div6 is 4, not 8.

Watch out

20−6+220-6+2 is 16, not 12. Adding first would insert brackets that were never there: 20−(6+2)20-(6+2).

Respect priority before left-to-right order.

Mixed exact calculations

The same order works with negative numbers, fractions and decimals. Decide the operation before choosing a convenient number form. A terminating decimal can become an exact fraction, avoiding an approximation when a different fraction has a recurring decimal.

Recall the sign rules with integers: (−3)×4=−12(-3)\times4=-12, (−3)×(−4)=12(-3)\times(-4)=12, 12÷(−4)=−312\div(-4)=-3 and (−12)÷(−4)=3(-12)\div(-4)=3. A product or quotient is positive when its two inputs have the same sign and negative when their signs differ. The same rules hold for signed fractions and decimals.

Worked example 5: exact decimals and fractions

Question. Work out (34+0.2)÷0.5\left(\frac34+0.2\right)\div0.5.

Plan. Convert the decimals to fractions, complete the bracket, then divide.

(34+0.2)÷0.5=(34+15)÷12=(1520+420)×2=1920×2=1910=1.9\begin{aligned} \left(\frac34+0.2\right)\div0.5 &=\left(\frac34+\frac15\right)\div\frac12\\ &=\left(\frac{15}{20}+\frac4{20}\right)\times2\\ &=\frac{19}{20}\times2\\ &=\frac{19}{10}=1.9 \end{aligned}

Answer. 1.91.9, exactly.

Check. Using exact decimals instead gives (0.75+0.20)÷0.5=0.95÷0.5=1.9(0.75+0.20)\div0.5=0.95\div0.5=1.9. Dividing by a half doubles the bracket's value.

The next fraction does not have a terminating decimal, and the multiplier is negative. Keep the calculation as fractions until the final result.

Worked example 6: avoiding premature rounding

Question. Work out (23+0.25)×(−1.2)\left(\frac23+0.25\right)\times(-1.2), giving an exact answer.

Plan. Use 0.25=140.25=\frac14 and −1.2=−65-1.2=-\frac65, then apply the negative sign to the product.

(23+0.25)×(−1.2)=(812+312)×(−65)=1112×(−65)=−1110=−1.1\begin{aligned} \left(\frac23+0.25\right)\times(-1.2) &=\left(\frac8{12}+\frac3{12}\right)\times\left(-\frac65\right)\\ &=\frac{11}{12}\times\left(-\frac65\right)\\ &=-\frac{11}{10}=-1.1 \end{aligned}

Answer. −1.1-1.1, exactly.

Check. Undo the multiplication: −1.1÷(−1.2)=1112-1.1\div(-1.2)=\frac{11}{12}. Subtracting 14\frac14 leaves 812=23\frac8{12}=\frac23, the original fraction. The positive bracket times a negative multiplier must give a negative result.

If you round 23\frac23 to 0.670.67 first, the product becomes −1.104-1.104, which is not the exact answer. On a calculator paper, enter the full fraction and the whole bracket, for example (2÷3+0.25)×(−1.2)(2\div3+0.25)\times(-1.2), and keep intermediate precision. A fraction display or multiplication-back check can confirm the hand result. On a non-calculator paper, the exact fraction method needs no electronic tool.

Problem-solving

Keep exact fractions when a decimal would repeat. Apply signs and operation order throughout, then convert or round only when the final demand requires it.

Complete the bracket exactly before dividing by the negative number.

Quick Recap

  • Complete brackets, then powers, then multiplication and division, then addition and subtraction.
  • Within either shared priority level, work from left to right.
  • The same sign rules apply to integers, fractions and decimals.
  • Keep exact values until any final rounding is required.

The opening expressions differ: 3+2×52=533+2\times5^2=53, while (3+2)×52=125(3+2)\times5^2=125.