KCSE 2025 Maths P1 Q1 — Order of Operations & Surds

KCSE 2025 Form 4 Numbers

Published

The Question

“Without using mathematical tables or a calculator, evaluate the square root of the expression eleven-twelfths minus one-third divided by one and one-half. Give the answer as a fraction in its simplest form.”

1

Read the expression and plan with BODMAS

The whole expression sits under one square root, so first simplify what is inside the root. Inside there is a subtraction and a division. Under the order of operations (BODMAS), division is handled before subtraction, so the first job is to work out one-third divided by one and one-half.

1112−13÷112\sqrt{\dfrac{11}{12} - \dfrac{1}{3} \div 1\tfrac{1}{2}}
2

Do the division first

Change the mixed number one and one-half into an improper fraction, three over two. Dividing by a fraction is the same as multiplying by its reciprocal, so flip three over two to become two over three. Multiply the numerators and multiply the denominators to get the result of the division.

112=321\tfrac{1}{2} = \dfrac{3}{2}
13÷32=13×23=29\dfrac{1}{3} \div \dfrac{3}{2} = \dfrac{1}{3} \times \dfrac{2}{3} = \dfrac{2}{9}
3

Subtract using a common denominator

Now subtract two-ninths from eleven-twelfths. The two fractions need the same denominator, so find the lowest common multiple of twelve and nine, which is thirty-six. Rewrite both fractions over thirty-six, then subtract the numerators.

1112−29=3336−836=2536\dfrac{11}{12} - \dfrac{2}{9} = \dfrac{33}{36} - \dfrac{8}{36} = \dfrac{25}{36}
4

Take the square root

The inside has simplified to twenty-five over thirty-six. Take the square root of the numerator and the square root of the denominator separately. Both are perfect squares, so the root comes out exactly with no surd left over.

2536=2536=56\sqrt{\dfrac{25}{36}} = \dfrac{\sqrt{25}}{\sqrt{36}} = \dfrac{5}{6}

Final Result

The value of the expression is five-sixths (5/6).

Why this method works

The method works because each operation is applied in the order BODMAS demands: division binds tighter than subtraction, so it must be resolved before the fractions can be combined. Turning the mixed number into an improper fraction lets us use the rule that dividing by a fraction equals multiplying by its reciprocal, which keeps everything as exact fractions rather than decimals. Rewriting over a common denominator is what makes the subtraction meaningful, since only like-sized parts can be combined. Finally, because the tidy result twenty-five over thirty-six has perfect squares on top and bottom, the square root of a fraction can be split into the root of the numerator over the root of the denominator, giving an exact answer without a calculator.

✓

Check: 5/6 squared is 25/36, which matches the value found inside the square root, confirming the answer.