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Changing the subject

Rearrange linear and non-linear formulas to change the subject, then substitute values to solve practical problems.

Worked examples

Make P the subject of I = Prt

Straightforward

Problem

The simple interest formula is I=PrtI = Prt, where II is interest earned, PP is the principal, rr is the annual interest rate and tt is time in years. Make PP the subject. Find PP when I=$180I = \$180, r=0.04r = 0.04 and t=3t = 3.

Make r the subject of A = pi r squared

Moderate

Problem

The area of a circle is A=πr2A = \pi r^2, where r>0r > 0. Make rr the subject. Find rr when A=153.94 cm2A = 153.94\,\text{cm}^2. Use π=3.14159\pi = 3.14159.

Make h the subject of SA = 2pi r(r + h)

Challenging

Problem

The total surface area of a closed cylinder is SA=2πr(r+h)SA = 2\pi r(r + h). Make hh the subject. Find hh when SA=376.99 cm2SA = 376.99\,\text{cm}^2 and r=4 cmr = 4\,\text{cm}. Use π=3.14159\pi = 3.14159.

Practise

Q1·Straightforward
The simple interest formula is I=PrtI = Prt. Make PP the subject. Find PP when I=450I = 450, r=0.05r = 0.05 and t=3t = 3.
Q2·Straightforward
The speed formula is v=dtv = \dfrac{d}{t}. Make dd the subject. Find dd when v=80 km/hv = 80\,\text{km/h} and t=2.5 ht = 2.5\,\text{h}.
Q3·Straightforward
The area of a rectangle is A=lwA = lw. Make ww the subject. Find ww when A=54 cm2A = 54\,\text{cm}^2 and l=9 cml = 9\,\text{cm}.
Q4·Straightforward
The circumference formula is C=2πrC = 2\pi r. Make rr the subject. Find rr when C=62.8C = 62.8 and π=3.14\pi = 3.14.
Q5·Moderate
The area of a circle is A=πr2A = \pi r^2, where r>0r > 0. Make rr the subject. Find rr when A=78.5A = 78.5 and π=3.14\pi = 3.14.
Q6·Moderate
The kinetic energy formula is E=12mv2E = \dfrac{1}{2}mv^2, where v>0v > 0. Make vv the subject. Find vv when E=400 JE = 400\,\text{J} and m=2 kgm = 2\,\text{kg}.
Q7·Moderate
The electrical power formula is P=V2RP = \dfrac{V^2}{R}, where V>0V > 0. Make VV the subject. Find VV when P=100 WP = 100\,\text{W} and R=4 ΩR = 4\,\Omega.
Q8·Moderate
The equation of motion v2=u2+2asv^2 = u^2 + 2as relates final velocity vv, initial velocity uu, acceleration aa and displacement ss. Make ss the subject. Find ss when v=20 m/sv = 20\,\text{m/s}, u=0u = 0 and a=4 m/s2a = 4\,\text{m/s}^2.
Q9·Challenging
The surface area of a sphere is SA=4πr2SA = 4\pi r^2, where r>0r > 0. Make rr the subject. Find rr when SA=200.96 cm2SA = 200.96\,\text{cm}^2 and π=3.14\pi = 3.14.
Q10·Challenging
The compound interest formula is A=P(1+r)nA = P(1+r)^n. Make PP the subject. Find PP when A=$2420A = \$2420, r=0.10r = 0.10 and n=2n = 2.
Q11·Challenging
Using v2=u2+2asv^2 = u^2 + 2as, make uu the subject where u>0u > 0. A car brakes from initial speed uu to rest (v=0v = 0) with deceleration a=−4 m/s2a = -4\,\text{m/s}^2 over s=50 ms = 50\,\text{m}. Find uu.
Q12·Challenging
The total surface area of a closed cylinder is SA=2πr(r+h)SA = 2\pi r(r + h). Make hh the subject. Find hh when SA=471.24 cm2SA = 471.24\,\text{cm}^2, r=5 cmr = 5\,\text{cm} and π=3.14159\pi = 3.14159.