Topic Exploration

Simple Equations & Variations

Master all past questions related to Simple Equations & Variations.

129 questions across all exams

Filter by exam

Direct Variations

9 Questions

The rate of consumption of petrol by a vehicle varies directly as the square of the distance covered. If 4 litres of petrol is consumed on a distance of 15 km, how far would the vehicle go on 9 litres of petrol?

22 ½ km

30 km

33 ¾km

45 km

The distance, d, through which a stone falls from rest varies directly as the square of the time, t, taken. If the stone falls 45 cm in 3 seconds, how far will it fall in 6 seconds?

90 cm

135 cm

180 cm

225 cm

G varies directly as the square of H. If G is 4 when H is 3, find H when G = 100.

225

In a particular year, the exchange rate of Naira (N) varies directly with the Dollar ( $). If N1122 is equivalent to$ 8, find the Naira equivalent of $36.

N8976

N5049

N140.25

N31.17

If N varies directly as M and N = 8 when M = 20, find M when N = 7.

$17^1/_2$

$18^1/_2$

Find the value P

If y varies directly as the square root of (x + 1) and y = 6 when x = 3, find x when y = 9.

If (x + 3) varies directly as y and x = 3 when y = 12, what is the value of x when y = 8?

-1

Find the value of k $\begin{pmatrix}X & 1 & 4 & P\\ Y & 0.5 & 1 & 2.5\end{pmatrix}$

0.5

1.0

1.5

0.0

Inverse Variations

12 Questions

If p α 1/q , which of the following is true?

$q\alpha p^2$

$q\alpha\frac{1}{p^2}$

q α√𝑝

q α 1/p

If x and y are variables and k is a constant, which of the following describes an inverse relationship between x and y?

y = kx

$y=\frac{𝑘}{x}$

y = √x

y = x + k

Given that y varies inversely as the square of x. If x = 3 when y = 100, find the equation connecting x and y

$yx^2=300$

$yx^2=900$

$y=\frac{100𝑥}{9}$

$y=900x^2$

If y varies inversely as $x^2$ , how does x vary with y?

x varies inversely as $y^2$

x varies inversely as √y

x varies directly as $y^2$

x varies directly as √y

If y α $\frac{1}{x^2}$ and y = $1\frac14$ when x = 4, find the value of y when x = $\frac12$

$2\frac12$

If y ∝ $\frac{1}{x^2}$ and x = 3 when y = 4, find y when x = 2

P varies inversely as the square of W. When W = 4, P = 9. Find the value of P when W = 9

$\frac{726}{16}$

$^{^{16}/_9}$

If x varies inversely as y and x = $\frac23$ when y = 9, find the value of y when x = $\frac34$

$\frac{1}{18}$

$\frac{8}{81}$

$\frac92$

P varies inversely as Q. The table below shows the value of Q for some selected values of P $\begin{pmatrix}P & 6 & 8 & 9 & 12\\ Q & 24 & 18 & ? & 12\end{pmatrix}$ What is the missing value of Q in the table?

If y α $\frac{1}{\surd𝑥}$ and x = 16 when y = 2, find x when y = 24

$^1/_9$

$^1/_6$

$^1/_3$

$^2/_3$

Given that p ∝ $\frac{1}{\surd r}$ and p = 3 when r = 16, find the value of r when p = $\frac32$

324

A variable P varies inversely as the square of Q. If P = 5 when Q = 6, find Q when P = 1.8.

Joint Variations

9 Questions

If x varies inversely as y and y varies directly as z, what is the relationship between x and z?

x ∝ z

$x\varpropto\frac{1}{z}$

$x\varpropto z^2$

$x\varpropto\frac{1}{Z^{}}$

If x varies inversely as y and y varies directly as z, what is the relationship between x and z

x ∝ z

$x\varpropto\frac{1}{z}$

$x\varpropto z^2$

$x\varpropto\frac{1}{Z^2}$

Given that p varies as the square of q and q varies inversely as the square root of r. How does p vary with r?

p varies as the square of r

p varies as the square root of r

p varies inversely as the square of r

p varies inversely as r

Given that p varies directly as q while q varies inversely as r, which of the following statements is true?

r varies directly as p

p varies inversely as r

p varies directly as r

q varies inversely as p

M varies directly as n and inversely as the square of p. If M = 3, when n = 2 and p = 1, find M in terms of n and p.

$M=\frac{2𝑛}{3p}$

$M=\frac{3𝑛^2}{2n^2}$

$M=\frac{𝑛^2}{2p}$

$M=\frac{3𝑛^{}}{2p^2}$

H varies directly as 𝑝 and inversely as the square of 𝑦. If H = 1, 𝑝 = 8 and 𝑦 = 2, find H in terms of 𝑝 and 𝑦.

$𝐻=\frac{𝑝}{4y^2}$

$𝐻=\frac{2𝑝}{y^2}$

$𝐻=\frac{𝑝}{2y^2}$

$𝐻=\frac{𝑝}{y^2}$

Given that R is directly proportional to L and inversely proportional to P, R = 3 when L = 9 and P = 0.8, find R when Z, = 15 and P = 1.8.

2.2

3.3

6.6

0.3

M varies directly as n and inversely as the square of p. If M= 3 when n = 2 and p = 1, find M in terms of n and p.

$M=\frac{3𝑛}{2p^2}$

$M=\frac{2𝑛}{3p^2}$

$M=\frac{2𝑛}{3p^{}}$

$M=\frac{3𝑛^2}{2p^2}$

M varies jointly as the square of n and the square root of q. If M= 24 when n = 2 and q = 4, find M when n = 5 and q = 9.

288

400

300

225

Partial Variations

4 Questions

If x ∝ (45 + ½ y), which of the following is true?

x varies directly as y

x varies inversely as y

x is partly constant and partly varies as y

x varies jointly as 45 and directly as y

Given that y = px + q and y = 5 when x = 3, while y = 4 when x = 2, find the values of p and q.

p = 1, q = 3

p = 1, q = 2

p = -2, q = 3

p = 3, q = -2

A variable W varies partly as M and partly inversely as P. Which of the following correctly 13 represents the relation with $𝒌^1$ and $𝒌^2$ as constants?

$W=\frac{k_1M}{k_2P}$

$W=(k_1+k_2)$ $\frac{M}{P}$

$W=k_1M+\frac{k_2}{P}$

$W=(k_1+k_2)M+P$

Make u the subject of the formula, $E=\frac{𝑚}{2𝑔}(v^2-u^2).$ .

$^{}u=\sqrt{v^2-\frac{2Eg}{m}}$

$^{}u=\sqrt{\frac{v^2}{m}-\frac{2Eg}{4}}$

$^{}u=\sqrt{v-\frac{2Eg}{m}}^{}$

$u=\sqrt{\frac{2v^2Eg}{m}}^{}$

Change of subject of a formula or relationship

25 Questions

Make k the subject of the relation $T=\sqrt{\frac{TK-H}{K-H}}$

$k=\frac{𝐻(𝑇2-1)}{T^2-T}$

$k=\frac{HT}{\left(T-1\right)^2}$

$k=\frac{H\left(T^2+1\right)}{T}$

$k=\frac{H\left(T^{}-1\right)}{T}$

If p = $\frac35\sqrt{\frac{q}{r}}$ , express q in terms of p and r

$\frac{9}{25}𝑝𝑟^2$

$\frac{9}{25}𝑝^2r$

$\frac{25}{9}𝑝^2r$

$\frac{25}{9}𝑝r^2$

Make s the subject of the relation: p = s + $\frac{𝑠𝑚^2}{nr}$

$s=\frac{𝑚𝑟𝑝}{𝑚𝑟+𝑚^2}$

$s=\frac{𝑚𝑟+𝑚^2}{nrp}$

$s=\frac{𝑛𝑟𝑝}{𝑚𝑟+𝑚^2}$

$s=\frac{𝑛𝑟𝑝}{𝑛𝑟+𝑚2}$

Make w the subject of the relation 𝑎 + 𝑏𝑐 𝑤𝑑 + 𝑓= 𝑔 $\frac{a+bc}{wd+f}=g$

$\frac{𝑎+bc-gf}{dg}$

$\frac{𝑎-bc+fg}{fg}$

$\frac{𝑎+bc-f}{dg}$

$\frac{𝑎+bc-dg}{dg}$

Make p the subject of the relation: $q=\frac{3𝑝}{r}+\frac{𝑠}{2}$

$p=\frac{2𝑝-rs}{6}$

p = 2pq - sr - 3

$\frac{2qr-s}{6}$

$\frac{2qr-rs}{6}$

Given that 𝒑⅓= $\frac{^3\surd q}{r}$ 𝑟make q the subject of the equation

q = p√𝑟

$q=p^3r$

$q=p^{}r^3$

$q=pr^{^1/_3}$

Make x the subject of the relation d = $\sqrt{\frac{6}{x}-\frac{y}{2}}$

$x=\frac{6}{d^2}+\frac{12}{y}$

$x=\frac{12}{2𝑑^2-𝑦}$

$x=\frac{12}{y}-2𝑑^2$

$x=\frac{12}{2𝑑^2+𝑦}^{}$

If y = $\sqrt{ax-b}$ express x in terms of y, a and b

$x=\frac{𝑦^2-b}{a}^{}$

$x=\frac{𝑦^{}+b}{a}^{}$

$x=\frac{𝑦^{}-b}{a}^{}$

$x=\frac{𝑦^2+b}{a}^{}$

Make f the subject of the relation v = u + ft.

$\frac{𝑣-u}{t}^{}$

$\frac{u-v}{t}^{}$

t(v + u)

$\frac{v}{u}^{}-t$

Make p the subject of the formula y = $\frac{𝑎+p}{a-p}$

$\frac{2𝑎-y}{a+y}^{}=p$

$\frac{𝑎y-1}{y+1}^{}=p$

$\frac{𝑎(𝑦-1)}{y+1}^{}=p$

$\frac{𝑎y}{y+1}^{}=p$

If P = $[\frac{Q\left(R-T\right)}{15}]^{\frac{^1}{3}}$

$T=\frac{𝑅+p^3}{15Q}^{}$

$T=\frac{𝑅-15p^3}{q}^{}$

$R-\frac{15𝑃^3}{Q}^{}$

$T-\frac{15R-Q}{p^3}$

Make q the subject of the relation t = $\sqrt{\frac{pq}{r}}-r^2q$

$q=\frac{𝑟𝑡^2}{p-r^3}$

$q=\frac{𝑡^2}{p-r^2}$

$q=\frac{r𝑡^{}}{p-r^3}$

q 𝑝−𝑟3 𝑟𝑡2

Make S the subject of the formula: $V=\frac{𝐾}{\surd𝑇-𝑆}$

$S=T-\frac{𝐾^2}{V^2}$

$S=\frac{𝐾^2-T}{V^2}$

$S=T-\frac{V^2}{K^2}$

$S=T-\frac{T\left(V^2-K^2\right)}{V^2}$

If h(m + n) = m(h + r) find h in terms of m, n and r

$h=\frac{𝑚𝑟}{2m+n}$

$h=\frac{𝑚𝑟}{n+m}$

$h=\frac{𝑚+r}{n}$

$h=\frac{𝑚r}{n}$

Make t the subject of the formula k = m $\sqrt{\frac{t-p}{r}}$

$\frac{𝑟k^2+p}{m^2}$

$\frac{𝑟k^2+^{pm^2}}{m^2}$

$\frac{𝑟k^2-p}{m^2}$

$\frac{𝑟k^2+p^2}{m^2}$

Given that p = x - $^1/x$ and q = $x^2+\frac{1}{x^2}$ , express q in the term of p

$(p^2+2)$

$(p^{}-2)^2$

$(p^{}+2)^2$

$(p^2-2)^{}$

If P = $\sqrt{QR\left(1+\frac{3t}{R}\right)}$ make R the subject of the formula

$R=\frac{3𝑄𝑇}{p^2-Q}$

$R=\frac{p^2-3T}{Q+1}$

$R=\frac{p^2-3T}{Q-1}$

$R=\frac{p^2-3QT}{p^2-Q}$

If $\frac{x}{a+1}+\frac{y}{b}$ = 1. Make y the subject of the relation

$\frac{a\left(b-x+1\right)}{b+1}$

$\frac{a+1}{b\left(a-x+1\right)}$

$\frac{b\left(a-x+1\right)}{a+1}$

$\frac{b}{a\left(b-x+1\right)}$

Make 𝑏 the subject of the relation 𝑙𝑏= $\frac12$ (𝑎+ 𝑏)ℎ

$\frac{𝑎ℎ}{2l-h}$

$\frac{2l-h}{al}$

$\frac{al}{2l-h}$

$\frac{al}{2-h}$

Make m the subject of the relation. $𝑘=\sqrt{\frac{m-y}{m+1}}$ .

$𝑚=\frac{𝑦+k^2}{k^2+1}$

$𝑚=\frac{𝑦+k^2}{1+k^2}$

$𝑚=\frac{𝑦-k^2}{k^2+1}$

$𝑚=\frac{𝑦-k^2}{1-k^2}$

Make U the subject of the relation: x = $\frac{2U-3}{3U+2}$

$U=\frac{2x+3}{3x-2}$

$U=\frac{2x-3}{3x-2}$

$U=\frac{2x+3}{2-3x}$

$U=\frac{2x+3}{3x+2}$

Make t the subject of k = 𝒎 $\sqrt{\frac{t-p}{r}}$ .

$t=\frac{rk^2+p}{m^2}$

$t=\frac{rk^2+pm^2}{m^2}$

$t=\frac{rk^2-p}{m^2}$

² $t=\frac{rk^2-p^2}{m^2}$

Make x the subject of the relation: y = $\frac{ax^3-b}{3z}$

$x=^3\sqrt{\frac{3yzb}{a}}$

$x=^3\sqrt{\frac{3yz+b}{a}}$

$x=^3\sqrt{\frac{3yz-b}{a}}$

$x=^3\sqrt{\frac{3y^3+b}{a}}$

Make 𝑹 the subject of the relation 𝑽= 𝝅𝒍 $(𝑹^2-𝒓^2).$ .

$R=^{}\sqrt{\frac{V}{\pi l}}+r^2$

$R=^{}\sqrt{\frac{V}{\pi l}}-r^2$

$R=^{}\sqrt{V-\pi lr^2}$

$R=^{}\sqrt{V+\pi lr^2}$

Make r the subject of the relation: $\frac{𝟏}{p}=\frac{𝒃}{t}+\frac{𝒄}{r}$

$r=\frac{pct}{t-pb}^{}$

$r=\frac{pct}{t+pb}^{}$

$r=\frac{pc}{t-pb}^{}$

$r=\frac{pc}{t+pb}^{}$

Substitution Of Algebraic Expression

13 Questions

Subtract (-y + 3x + 5z) from (4y - x - 2z).

5y - 4x - 7z

3y + 2x + 3z

-5y + 4x + 7z

2x - 5y + 3z

Given that x = $\frac{𝟏}{2}$ and y = 4, evaluate $3x^2y+xy^2$

-5

-1

Given that p = 2, q = -5 and r = - 4, evaluate $3p^2-q^2-r^3$

101

-27

If x = 3 and y = —I. evaluate 2 $(x^2-y^3)$ .

If a = 3 and b = -7, find the value of $\frac{5𝑏+\left(a+b\right)^2}{\left(a-b\right)^2}$ .

0.51

0.19

-0.19

-0.51

If E = $\frac{𝑀N}{S+N}$ and E = 75, M = 120, N= 5000, find S.

1000

2000

3000

4000

If F = $\frac95$ C + 32, find C when F = 98.6

If x = 3, y = 2 and z = 4, what is the value of $3x^2$ - 2y+z?

Given that P = $x^2$ + 4x - 2, Q = 2x -1 and Q -P =2, find x

-2

-1

If 2x + y = 10, and y ≠ 0, which of the following is not a possible value of x?

Evaluate $\frac{𝑥^2+x-2}{2x^2+x-3}$ when x = -1

-2

-1

-½

Given that one of the roots of the equation $2𝑥^2$ + (k + 2)x + k = 0 is 2, find the value of k.

-4

-2

-1

$-\frac14^{}$

If (x + 2) is a factor of $𝑥^2$ + Px - 10, find the value of P.

-3

-7

Simple Equations

4 Questions

If x $kmh^{-1}$ =y $ms^{-1}$ , the y =

$^7/_{18x}$

$^{11}/_{20x}$

$^4/_{15x}$

$^5/_{18x}$

If 3y = 243, find the value of y.

If 30% of y is equal to x, what, in terms of x, is 30% of 3y?

$\frac{x}{9}$

$\frac{x}{3}$

$x$

$3x$

If (¼)(2-y) = 1, find y.

-2

-1/2

1/2

Word Problems Leading To Simple Equations

11 Questions

The square root of a number is 2k. What is half of the number

$\surd\frac{k}{2}$

$\surd k$

½ $k^2$

$2k^2$

A car travels at x km per hour for 1 hour and at y km per hour for 2 hours. Find its average speed.

$\frac{2𝑥+2y}{3}kmh^{-1}$

$\frac{𝑥+y}{3}kmh^{-1}$

$\frac{𝑥+2y}{3}kmh^{-1}$

$\frac{2𝑥+y}{3}kmh^{-1}$

A train moving at a uniform speed, covers 36 km in 21 minutes. How long does it take to cover 60 km?

35 minutes

40 minutes

45 minutes

90 minutes

The temperature in a chemical plant was -5°C at 200 a.m. The temperature fell by 6°C and then rose again by 7°C. What was the final temperature?

-6°C

-4°C

4°C

8°C

A car moves at an average speed of $30kmh^{-1}$ , how long does it take to cover 200 metres?

2.4 sec

24 sec

144 sec

240 sec

A car is travelling at an average speed of 80km/ hr. Its speed in metres per second (m/s) is

13.3m/s

22.2m/s

133.3m/s

222.2m/s

To arrive on schedule, a train is to cover a distance of 60km at 72km/hr. If it starts 10 minute late, at what speed must it move to arrive on schedule?

60km/hr

80km/hr

90km/hr

108km/hr

A boy walks 800 m in 20 minutes. Calculate his average speed in km per hour

2.4

What must be added to (2x - 3y) to get (x -2y)?

5y - x

y - x

x - 5x

x - y

A man bought 220 mangoes at N5x. He sold each for 3x kobo and made a gain of N8. Find the value of x.

Two buses start from the same station at 9:00 am. and travel in opposite directions along the same straight road. The first bus travels at a speed of 72 km/h and the second at 48 km/h. At what time will they be 240 km apart?

1:00 pm

12:00 noon

11:00 am

10:00 am

Linear Equations

11 Questions

Solve: 6(x - 4) + 3(x + 7) = 3

$\frac32$

$\frac23$

$\frac12$

$\frac13$

Solve the equation $\frac{𝑚}{3}+\frac12=\frac34+\frac{𝑚}{4}$

-3

-2

Find the value of x in 0.5x + 2.6 = 5x + 0.35

0.5

2.6

Solve for t in the equation ¾ t + ⅓ (21 - t) = 11

$^9/_{14}$

$^9/_{13}$

$9^3/_5$

Solve the equation $\frac{4}{a}+\frac{1}{5a}=3$

$2\frac15$

$1\frac25$

$1\frac13$

$1\frac{4}{15}$

Given that $\frac{6𝑥-y}{x+2y}$ = 2, find the value of 𝑥 𝑦

$\frac38$

$\frac58$

$\frac45$

$\frac54$

If $\frac{2}{\left(x-3\right)}-\frac{3}{\left(x-2\right)}=\frac{p}{\left(x-3\right)\left(x-2\right)}$ ,find p.

(5 —x)

(x + 5)

(13 —x)

(5 x - 13)

Solve: 1 + $^3\sqrt{x-3}$ = 4.

Solve for x in the equation $^3/_5$ (2x -1) = $^1/_4$ (5x- 3)

If x + y = 2y - x + 1 = 5, find the value of x.

-1

Solve the equation: $^2/_3$ (x + 5) = $^1/_4$ (5x - 3)

$1^1/_7$

$1^8/_{23}$

Linear Equations Involving Fraction

6 Questions

Find the value of x such that the expression $\frac{1}{x}+\frac{4}{3x}-\frac{5}{6x}+1$ equals zero.

$\frac16$

$\frac14$

$\frac{-3}{2}$

$\frac{-7}{6}$

Solve the equation: $\frac{2y-1}{3}-\frac{3y-1}{4}$ = 1

−8

−13

Find the value of x such that $\frac{1}{x}+\frac{4}{3x}-\frac{5}{6x}$ + 1 is zero.

$\frac16$

$\frac14$

$\frac{-3}{2}$

$\frac{-7}{6}$

If $\frac{2}{x-3}-\frac{3}{x-2}$ is equal to $\frac{p}{\left(x-3\right)\left(x-2\right)}$ , Find P.

-x-5

-(x + 3).

5x - 13

5-x

Solve the equation: $\frac15+\frac{1}{x}=3$

$\frac15$

$\frac25$

$\frac35$

$\frac45$

Solve: $\frac{y+1}{2}-\frac{2y-1}{3}$ = 4.

𝑦= 19

𝑦= −19

𝑦= −29

𝑦= 29

Simultaneous Linear Equations

25 Questions

Given that 2p - m = 6 and 2p + 4m = 1, find the value of (4p + 3m).

Solve for y in the equations: 2x − 3y = 22 ; 3x + 2y = 7

−5

−4

If 3p - q = 6 and 2p + 3q = 4, find q

$\frac12$

$\frac23$

Solve the following simultaneous equations: x + y = $^3/_2$ ; x - y = $^5/_2$ and use your result to find the value of 2y + x

-2

-1

$^1/_2$

If x + y = 12 and 3 x -y = 20, find the value of 2x - y.

Solve the equations: 4x - y = 11, 5x + 2y = 4

x = 6, y = 13

x = -2, y = -3

x = -2, y = 3

x = 2, y = -3

Find (x - y) if 4x - 3y = 7 and 3x - 2y = 5.

-3

-2

If 2x + y = 7 and 3x - 2y = 3, by how much is 7x greater than 10?

Given that x + y = 7 and 3x - y = 5, evaluate y/2 -3

-1

Solve the simultaneous equations y = 3x; 4y- 5x = 14

-2,-6

2, 6

2,-6

-2, 6

Solve 3x - 2y = 10 and x + 3y = 7 simultaneously.

x = -4 and y = 1

x = -1 and y = —4

x = 1 and y = 4

x = 4 and y = 1

If 4x + 2y = 16 and 6x - 2y = 4, find the value of (y - x).

Find the value of p if $\frac{𝟏}{4}$ 𝒑+ 𝟑𝒒= 𝟏𝟎 𝒂𝒏𝒅 𝟐𝒑− $\frac{𝟏}{3}$ q = 7.

-3

-4

If p - 2g + 1 = g + 3p and p - 2 = 0, find g.

-2

-1

Given that 2x + y = 7 and 3x - 2y = 3, by how much is 7x greater than 10?

Solve the equations: 3x - 2y = 7, x + 2y = -3

x = 1, y = -2

x = 1, y = 3

x = 2, y = -1

x = 4, y = -3

Solve the simultaneous equations: x + y= 2 and 3x - 2y = 1.

x = 2 and y = 1

x = 1 and y = 1

x = 1 and y = 2

x = -1 and y = 1

Solve the following simultaneous equations: 2x + 3y = 7, x + 5y = 0.

x = 5, y = -1

x = $\frac15$ , y = -1

x = - $\frac15$ , y =1

x = -5, y = 1

If x + 0.4y = 3 and y = 1/2 x, find the value of (x + y).

$1^1/_4$

$2^1/_2$

$3^3/_4$

If $\frac{𝟏}{2}$ x + 2y = 3 and 3 2x - 2y = 1, find (x + y)

lf y + 2x = 4 and y - 3x = -1, find the value of (x + y)

-1

If 3𝑝 = 4𝑞 and 9𝑝 = 8𝑞 – 12, find the value of 𝑝𝑞.

-7

-12

If 2x - 3y = -11 and 3x + 2y = 3, evaluate (y - x)2.

If 𝟑𝒙− 𝟐𝒚 = −𝟓 and 𝒙 + 𝟐𝒚 = 𝟗, find the value of $\frac{𝒙-y}{x+y}$ .

$\frac{𝟓}{3}$

$\frac35$

$-\frac35$

$-\frac53$

Given that 𝒙+ 𝒚= 1 and 𝒙+ 𝟑𝒚 = 5, find the value of ( $𝒙^2$ + 𝟒𝒙𝒚+ $3y^2$ ).

1 2 3