Variation

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1. 1Direct, inverse and joint variationWriting 'y varies as…' as an equation with a constant k, finding k from one pair of values, and using the equation: direct, inverse, square, square-root and joint variation.Done16 min
2. 2Partial variation and proportion problemsQuantities that are partly constant and partly vary, solved with two simultaneous equations; and everyday proportion problems such as workers and days.Done15 min

Past questions (52)

From 1983–2025: 17 WAEC, 7 NECO, 28 JAMB. Theory questions first, then objective, newest first.

Theory (14)

• WAEC 2025 · Paper 2 · Q2$y$ is partly constant and partly varies inversely as the square of $x$.
• NECO 2024 · Paper 2 · Q7Solve the equation $\dfrac{19}{12} - \dfrac{3}{4(x - 4)} = \dfrac56$.
• WAEC 2024 · Paper 2 · Q1The time ($t$) taken to buy fuel at a filling station varies directly as the number of vehicles ($V$) in a queue and inversely as the number …
• NECO 2023 · Paper 2 · Q1$P$ varies directly as the square of $Q$ and inversely as the cube of $Z$. When $P = 5$, $Q = 3$ and $Z = 1$. Find:
• NECO 2022 · Paper 2 · Q9Make $M$ the subject of the formula $M = \sqrt{\frac{N - M^2 f + 1}{d - N}}$. $M =$
• WAEC 2021 · Paper 2 · Q8The time ($T$) spent on erecting a concrete pillar is partly constant and partly varies as three times the base area ($A$). If it took 27 …
• WAEC 2019 · Paper 2 · Q6The force of attraction, $F$, between two bodies varies directly as the product of their masses, $m_1$ and $m_2$, and inversely as the square …
• WAEC 2019 · Paper 2 · Q10A twenty-kilogram bag of rice is consumed by $m$ boys in 10 days. When four more boys joined them, the same quantity of rice lasted only 8 days. …
• WAEC 2019 · Paper 2 · Q5The cost of dinner for a group of tourists is partly constant and partly varies as the number of tourists present. It costs $740.00 when 20 tourists were present and $960.00 …
• WAEC 2017 · Paper 2 · Q11It takes 8 students two-thirds of an hour to fill 12 tanks with water. How many tanks of water will 4 students fill in one-third of an hour at …
• WAEC 2013 · Paper 2 · Q13Given that $Y = \{x : -2 \le x \le 5\}$ and $W = \{x : 1 < x < 6\}$, illustrate $Y \cap W$ on the number line.
• WAEC 2011 · Paper 2 · Q13$P$ varies directly as $Q$ and inversely as the square of $R$. If $P = 1$ when $Q = 8$ and $R = 2$, find the value of $Q$ when $P = 3$ and $R = 5$.
• WAEC 2010 · Paper 2 · Q13The third term of a Geometric Progression (G.P.) is 24 and its seventh term is $4\frac{20}{27}$. Find its first term.
• WAEC 2008 · Paper 2 · Q6If $2^{x + y} = 16$ and $4^{x - y} = \frac{1}{32}$, find the values of $x$ and $y$.

Objective (38)

• WAEC 2025 · Paper 1 · Q15If $y \propto \dfrac{1}{x^2}$ and $x = 3$ when $y = 4$, find $y$ when $x = 2$.
• WAEC 2024 · Paper 1 · Q13A variable $W$ varies partly as $M$ and partly inversely as $P$. Which of the following correctly represents the relation with $k_1$ and …
• NECO 2023 · Paper 1 · Q17A helicopter takes 3 hours from Kano to Lagos at a constant speed. How long does the same journey take another helicopter at a quarter of the …
• NECO 2023 · Paper 1 · Q19$y$ is partly constant and partly varies as $x$. When $y = 2$, $x = 3$ and when $y = 5$, $x = 6$. Find the relationship between $x$ and $y$.
• WAEC 2023 · Paper 1 · Q10$M$ 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 $q = 9$.
• NECO 2022 · Paper 1 · Q32$A$ varies inversely as the square of $B$ and $B$ varies directly as the square of $C$. Find the equation connecting $A$ and $C$, where $K$ is …
• NECO 2022 · Paper 1 · Q33If $M$ varies inversely as $N$ and $M = \frac{15}{2}$ when $N = 4$, find the value of $N$ when $M = \left(\frac{1}{20}\right)^{-1}$.
• WAEC 2022 · Paper 1 · Q9$M$ 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 $p$.
• WAEC 2020 · Paper 1 · Q16Given that $x$ is directly proportional to $y$ and inversely proportional to $z$, and $x = 15$ when $y = 10$ and $z = 4$, find the equation connecting …
• WAEC 2019 · Paper 1 · Q12$H$ 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 $y$.
• JAMB 2018 · UTME · Q38A baking recipe calls for $2.5\text{ kg}$ of sugar and $4.5\text{ kg}$ of flour. With this recipe some cakes were baked using $24.5\text{ kg}$ …
• JAMB 2017 · UTME (set 2) · Q25$y$ is inversely proportional to $x$, and $y = 6$ when $x = 7$. Find the constant of the variation.
• JAMB 2017 · UTME (set 2) · Q8If temperature $t$ is directly proportional to heat $h$, and when $t = 20^\circ\text{C}$, $h = 50\text{ J}$, find $t$ when $h = 60\text{ J}$.
• JAMB 2014 · UTME · Q15$y$ varies directly as $w^2$. When $y = 8$, $w = 2$. Find $y$ when $w = 3$.
• JAMB 2014 · UTME · Q16$P$ varies directly as $Q$ and inversely as $R$. When $Q = 36$ and $R = 16$, $P = 27$. Find the relation between $P$, $Q$ and $R$.
• JAMB 2013 · UTME · Q14$P$ varies jointly as $m$ and $u$, and inversely as $q$. Given that $P = 4$, $m = 3$, $u = 2$ and $q = 1$, find the value of $P$ when $m = 6$, …
• JAMB 2013 · UTME · Q15If $r$ varies inversely as the square root of $s$ and $t$, how does $s$ vary with $r$ and $t$?
• JAMB 2012 · UTME · Q14If $y$ varies directly as $\sqrt n$ and $y = 4$ when $n = 4$, find $y$ when $n = 1\frac79$.
• JAMB 2012 · UTME · Q15$U$ is inversely proportional to the cube of $V$, and $U = 81$ when $V = 2$. Find $U$ when $V = 3$.
• JAMB 2011 · UTME · Q16If $x$ varies directly as the square root of $y$ and $x = 81$ when $y = 9$, find $x$ when $y = 1\frac79$.
• JAMB 2011 · UTME · Q17$T$ varies inversely as the cube of $R$. When $R = 3$, $T = \frac{2}{81}$. Find $T$ when $R = 2$.
• JAMB 2010 · UTME · Q17If $y$ varies directly as the square root of $x$ and $y = 3$ when $x = 16$, calculate $y$ when $x = 64$.
• JAMB 2010 · UTME · Q18If $x$ is inversely proportional to $y$ and $x = 2\frac12$ when $y = 2$, find $x$ if $y = 4$.
• JAMB 2004 · UME · Q14$y$ is inversely proportional to $x$ and $y = 4$ when $x = \frac12$. Find $x$ when $y = 10$.
• JAMB 2004 · UME · Q15The length $L$ of a simple pendulum varies directly as the square of its period $T$. If a pendulum with period 4 s is 64 cm long, find the length …
• JAMB 2003 · UME · Q19$x$ varies directly as the product of $U$ and $V$ and inversely as their sum. If $x = 3$ when $U = 3$ and $V = 1$, what is the value of $x$ if …
• JAMB 2003 · UME · Q22The length a person can jump is inversely proportional to his weight. If a 20 kg person can jump 1.5 m, find the constant of proportionality.
• JAMB 2002 · UME · Q33The time taken to do a piece of work is inversely proportional to the number of men employed. If it takes 45 men 5 days to do the work, how long …
• JAMB 2002 · UME · Q42If $x$ varies directly as $\sqrt n$ and $x = 9$ when $n = 9$, find $x$ when $n = \frac{17}{9}$.
• JAMB 1991 · UME · Q15If $w$ varies inversely as $\frac{uv}{u + v}$ and $w = 8$ when $u = 2$ and $v = 6$, find a relationship between $u$, $v$ and $w$.
• JAMB 1989 · UME · Q15The cost of dinner for a group of students is partly constant and partly varies directly as the number of students. If the cost is ₦74.00 for …
• JAMB 1988 · UME · Q14If $x$ varies inversely as the cube root of $y$, and $x = 1$ when $y = 8$, find $y$ when $x = 3$.
• JAMB 1987 · UME · Q19If $P$ varies inversely as $V$ and $V$ varies directly as $R^2$, find the relationship between $P$ and $R$, given that $R = 7$ when $P = 2$.
• JAMB 1986 · UME · Q16If $x$ varies directly as $y^3$, and $x = 2$ when $y = 1$, find $x$ when $y = 5$.
• JAMB 1985 · UME · Q16In a restaurant, the cost of providing a particular type of food is partly constant and partly inversely proportional to the number of people. …
• JAMB 1983 · UME · Q14$y$ varies partly as the square of $x$ and partly as the inverse of the square root of $x$. Write down the expression for $y$ if $y = 2$ when …
• JAMB 1983 · UME · Q23If $w$ varies inversely as $V$ and $u$ varies directly as $w^3$, find the relationship between $u$ and $V$, given that $u = 1$ when $V = 2$.
• JAMB 1983 · UME · Q6If $x$ is jointly proportional to the cube of $y$ and the fourth power of $z$, in what ratio is $x$ increased or decreased when $y$ is halved …