KCSE 2025 Maths P2 Q18 — Rational Equation from Club Registration

KCSE 2025 Form 4 Algebra

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The Question

“Registering for a Music club costs Ksh 7,000 and for a Mathematics club costs Ksh 7,200. Each cost is shared equally among that club's members. Both clubs originally had x members each, but before payment 5 Music club members switched to the Mathematics club. (a)(i) Write an expression in x for the amount each Music club member pays after the switch. (a)(ii) Write an expression in x for the amount each Mathematics club member pays after the switch. (b)(i) Given that a Music club member contributed Ksh 40 more than a Mathematics club member, form an equation in x and determine the value of x. (b)(ii) Determine the percentage increase in the amount contributed by each remaining Music club member due to the switch.”

1

Write the two payment expressions (a)(i) and (a)(ii)

Each member's payment is the total cost shared equally, so divide the total by the number of members. After the switch the Music club has lost 5 members and has x minus 5, while the Mathematics club has gained them and has x plus 5. Divide each club's fixed cost by its new membership.

Music: 7000x−5\text{Music: } \frac{7000}{x - 5}
Maths: 7200x+5\text{Maths: } \frac{7200}{x + 5}
2

Form the equation (b)(i)

The condition says a Music member paid Ksh 40 more than a Mathematics member, so the Music payment minus the Mathematics payment equals 40. Write that as an equation in x using the two expressions above.

7000x−5−7200x+5=40\frac{7000}{x - 5} - \frac{7200}{x + 5} = 40
3

Clear the fractions

To remove the denominators, multiply every term by the common denominator (x - 5)(x + 5). This cancels the fractions on the left and, because (x - 5)(x + 5) is a difference of two squares, gives x squared minus 25 on the right.

7000(x+5)−7200(x−5)=40(x−5)(x+5)7000(x + 5) - 7200(x - 5) = 40(x - 5)(x + 5)
7000x+35 000−7200x+36 000=40(x2−25)7000x + 35\,000 - 7200x + 36\,000 = 40(x^{2} - 25)
4

Simplify to a quadratic (b)(i)

Collect like terms on the left and expand the right. Then bring every term to one side so the equation equals zero, and divide through by 40 to get a simpler quadratic with small coefficients.

−200x+71 000=40x2−1 000-200x + 71\,000 = 40x^{2} - 1\,000
40x2+200x−72 000=040x^{2} + 200x - 72\,000 = 0
x2+5x−1 800=0x^{2} + 5x - 1\,800 = 0
5

Solve the quadratic (b)(i)

Factorise by finding two numbers that multiply to -1,800 and add to +5; those are -40 and +45. Set each factor to zero. A club cannot have a negative number of members, so reject the negative root and keep the positive value of x.

(x−40)(x+45)=0(x - 40)(x + 45) = 0
x=40orx=−45  (rejected)x = 40 \quad \text{or} \quad x = -45 \; (\text{rejected})
x=40x = 40
6

Find the percentage increase (b)(ii)

Before the switch the Music club had 40 members, so each paid 7,000 divided by 40. After 5 left, 35 members remained and each paid 7,000 divided by 35. The percentage increase is the rise in payment divided by the original payment, times 100.

Before: 700040=175,After: 700035=200\text{Before: } \frac{7000}{40} = 175, \qquad \text{After: } \frac{7000}{35} = 200
% increase=200−175175×100=25175×100≈14.29%\%\,\text{increase} = \frac{200 - 175}{175} \times 100 = \frac{25}{175} \times 100 \approx 14.29\%

Final Result

Each Music member pays 7000/(x - 5) and each Mathematics member pays 7200/(x + 5). Solving 7000/(x - 5) - 7200/(x + 5) = 40 gives x squared + 5x - 1800 = 0, so x = 40. The amount each remaining Music member pays rises from Ksh 175 to Ksh 200, a percentage increase of about 14.29%.

Why this method works

Sharing a fixed cost equally makes each payment a fraction with the number of members in the denominator, so fewer members means a larger share — this is why the Music payment climbs when 5 people leave. Multiplying through by the common denominator is valid because it scales both sides of a true equation equally, turning an awkward fractional equation into an ordinary quadratic that factorises. Rejecting the negative root reflects the real-world constraint that a membership count must be a positive whole number.

✓

With x = 40: Music pays 7000/35 = 200 and Maths pays 7200/45 = 160, and 200 - 160 = 40, exactly the stated difference.