Alliance Form 3 P1 Q11 — Simplifying Algebraic Fractions

KCSE (Form 3) Form 3 Algebra

Published

The Question

“Simplify the expression formed by adding the fraction A over 2(A + B) to the fraction B over 2(A - B), giving your answer as a single simplified algebraic fraction.”

1

Identify the denominators

Look at the two fractions you are adding. The first has denominator two times the bracket A plus B, and the second has denominator two times the bracket A minus B. To add fractions you must first give them the same bottom, so you need a shared common denominator that both of these divide into.

A2(A+B)+B2(A−B)\frac{A}{2(A+B)} + \frac{B}{2(A-B)}
2

Build the common denominator

The lowest common denominator must contain the factor 2 once and both bracket factors, A plus B and A minus B. Multiplying these together gives the single denominator that both original fractions can be rewritten over without leaving anything out.

2(A+B)(A−B)2(A+B)(A-B)
3

Rewrite each numerator

To move the first fraction onto the common denominator you multiply its top and bottom by the factor it is missing, which is A minus B, so its numerator becomes A times the bracket A minus B. The second fraction is missing the factor A plus B, so its numerator becomes B times the bracket A plus B.

A(A−B)+B(A+B)2(A+B)(A−B)\frac{A(A-B) + B(A+B)}{2(A+B)(A-B)}
4

Expand the numerator

Now multiply out each bracket on the top. A times the bracket A minus B gives A squared minus AB, and B times the bracket A plus B gives AB plus B squared. Keep the denominator as it is for now.

A(A−B)=A2−ABA(A-B) = A^{2} - AB
B(A+B)=AB+B2B(A+B) = AB + B^{2}
A2−AB+AB+B22(A+B)(A−B)\frac{A^{2} - AB + AB + B^{2}}{2(A+B)(A-B)}
5

Cancel and simplify

On the top the minus AB and the plus AB are opposites, so they cancel to leave A squared plus B squared. On the bottom, the product of A plus B and A minus B is the difference of two squares, which is A squared minus B squared. This gives the final simplified fraction.

A2+B22(A2−B2)\frac{A^{2} + B^{2}}{2(A^{2} - B^{2})}

A plus B times A minus B is a difference of two squares, equal to A squared minus B squared.

Final Result

The expression simplifies to (A squared plus B squared) divided by 2 times (A squared minus B squared).

Why this method works

The method works because fractions can only be added once they share a denominator, and the smallest safe choice is a denominator built from every distinct factor of the two bottoms. Rewriting each fraction over that common denominator does not change its value, since you multiply the top and bottom by the same factor. Once the tops are combined, the equal and opposite middle terms cancel, and recognising the bottom as a difference of two squares keeps the answer in its most compact factored form.

✓

Substitute simple numbers, say A = 2 and B = 1: the original gives 2/6 + 1/2 = 5/6, and the answer gives 5/6, confirming the result.