$A$ is directly proportional to $B$. If $A = 20$ when $B = 4$,
- find $A$ when $B=3$.
- find $B$ when $A=10$.
$A$ is directly proportional to $B$. If $A = 20$ when $B = 4$,
This method, while it uses simple unitary method working, misses out a vital step which is amazingly useful in this work: namely, we don't get a working formula out of it, though we can still answer the questions.
It is important to understand that while this method may be easier to understand and use, you must also learn the algebraic (more formal) approach as well as often you will also be asked for the formula.
If you only wish to learn a single method at this stage, you must make it the formal algebraic method.
$A$ is directly proportional to $B$. If $A = 20$ when $B = 4$,
Translate the proportional relationship to an formula with an unknown constant.
What this squiggly fish symbol means is that there exists a constant (number) which multiplies by $B$ to give $A$, and vice versa, so it can be re-written as $A = kB$, where $k$ is the unknown constant of proportionality.
$x$ is directly proportional to $y$. If $x = 14$ when $y = 2$,
The height $H$ mm, of a pile of sheets of A4 paper is directly proportional to the number of sheets $N$, in the pile. A pile of sheets whose height is $28$ mm contains $250$ sheets of paper.
Remember to put the units back in when you state your answer. If you are not sure why units are so important, please go onto YouTube and watch the Stonehenge section of the film This is Spinal Tap
. Or preferably watch all of it, as it is a masterpiece.
This exercise is designed purely to give practice in very similar examples. The idea, here, is to get completely on top of the technique, before we start to look at more involved wordy problems. Do try to notice that despite the fact that the variables change from question to question, the technique used is identical in every other regard.
$P$ is directly proportional to $Q$. When $P = 3$, $Q = 5$.
$A$ is directly proportional to $B$. When $A = 3$, $B = 8$.
$x$ is directly proportional to $y$. When $x =7$, $y =2$.
$m$ is directly proportional to $n$. When $m = 270$, $n = 5$.
$p$ is directly proportional to $q$. When $p = 15.36$, $q = 4.8$.
$F$ is directly proportional to $a$. When $F = 35.075$, $a = 5.8$.
$a$ is directly proportional to $b$. When $a = 15$, $b = 36$.
$i$ is directly proportional to $k$. When $i = 3$, $k = 7$.
$v$ is directly proportional to $t$. When $v = 1.5$, $t = 5.6$.
$g$ is directly proportional to $h$. When $g = 6.5$, $h = 240$.
Read the questions with great care. Lay out your working with equal care. Check your answers.
The distance traveled by a car ($S$ km) is directly proportional to the amount of petrol ($P$ litres) used. The car travels $378$ km on $7$ litres of petrol.
The distance traveled by a train is directly proportional to the time taken for the journey. The train travels $105$ miles in $3$ hours.
The numbers of children who can safely use a municipal playground is directly proportional to the area of the playground. It is known that $91$ children can safely use a playground with an area of $26\text{m}^2$.
The time taken for a pan of water to boil is directly proportional to the quantity of water in the pan. It is known that $2.4$ litres of water will boil in $150$ seconds.
The number of spaces in a car park varies as the floor area of the car park. The equation of proportionality is $S = \frac{1}{15}A$, where $S$ represents the number of spaces and $A$ is the area in square metres. An additional $750\text{m}^2$ is added to the car park. How many extra spaces will this add to the car park?
$50$ additional spaces.
$2$ ice-creams are eaten by $2$ children in $2$ minutes. How long will it take $5$ children to eat $5$ ice-creams?
How long does it take for a child to eat an ice-cream (according to this puzzle)?
$2$ minutes
If it takes $2$ minutes for the $2$ children to eat their ice-creams, then it takes one of the children $2$ minutes to eat their ice-cream. So it will still take $2$ minutes if one million children are eating one million ice-creams, if we assume that they eat one each, at exactly the same rate.