Alliance Form 3 P1 Q1 — Order of Operations (BODMAS)

KCSE (Form 3) Form 3 Numbers

Published

The Question

“Without using a calculator, evaluate the fraction whose numerator is negative two times the bracket five plus three, minus nine divided by three, plus five, and whose denominator is negative three times negative five, plus negative two times four.”

1

Set up the plan with BODMAS

The expression is a fraction, and the fraction bar means divide. So first simplify the numerator (the top) completely, then simplify the denominator (the bottom) completely, and only divide the two results at the very end. Within the top and bottom, follow BODMAS: brackets first, then orders (powers), then division and multiplication, and finally addition and subtraction.

−2×(5+3)−9÷3+5(−3×−5)+(−2×4)\frac{-2 \times (5+3) - 9 \div 3 + 5}{(-3 \times -5) + (-2 \times 4)}
2

Simplify the numerator

Start inside the bracket, since brackets come first: five plus three is eight. Next handle multiplication and division before any adding or subtracting. Negative two times eight is negative sixteen, and nine divided by three is three. The top now reads negative sixteen, minus three, plus five. Working left to right, negative sixteen minus three is negative nineteen, and negative nineteen plus five is negative fourteen.

−2×(5+3)=−2×8=−16-2 \times (5+3) = -2 \times 8 = -16
9÷3=39 \div 3 = 3
−16−3+5=−19+5=−14-16 - 3 + 5 = -19 + 5 = -14
3

Simplify the denominator

Multiplication comes before addition here too. Negative three times negative five is positive fifteen, because two negatives multiplied together give a positive. Negative two times four is negative eight, because a negative times a positive gives a negative. The bottom becomes fifteen plus negative eight, and adding a negative is the same as subtracting, so this is fifteen minus eight, which is seven.

−3×−5=15-3 \times -5 = 15
−2×4=−8-2 \times 4 = -8
15+(−8)=15−8=715 + (-8) = 15 - 8 = 7
4

Divide the top by the bottom

Now carry out the division that the fraction bar represents: negative fourteen divided by seven. Fourteen divided by seven is two, and a negative divided by a positive stays negative, so the result is negative two.

−147=−2\frac{-14}{7} = -2

Final Result

The expression evaluates to -2.

Why this method works

BODMAS is not an arbitrary rule; it fixes a single agreed order so that everyone reads the same expression the same way and reaches the same value. Brackets are done first because they group a calculation that must be treated as one quantity, and multiplication and division bind more tightly than addition and subtraction. Treating the fraction bar as a grouping that divides the whole top by the whole bottom is what lets us safely simplify each part in isolation before combining them. The sign rules follow from the meaning of negatives: two negatives multiplied reverse direction twice and land positive, while adding a negative moves in the subtracting direction.

✓

Recompute directly: top -2(8) - 3 + 5 = -14, bottom 15 - 8 = 7, and -14 divided by 7 = -2.