4.2 Measuring-numbers

Up until now, we’ve only been talking about multiplication with counting-numbers; now we’re going to be dealing with measuring-numbers as well. Try answering these questions.

Question 4.2.1
  1. If I can walk $100$ metres in $1$ minute, how far can I walk in $7$ minutes? Show answer

  2. One minute is $60$ seconds; how many minutes is $720$ seconds? Show answer

Even though these questions involve situations we haven’t dealt with yet, you can answer them with the skills you’ve already learnt, but what if I changed the numbers? I could have asked how far I can walk in $6.2$ minutes, and might only be able to walk $85.1$ metres in a minute. You can’t think of that as $6.2$ groups of $85.1$; that doesn’t make any sense. I could have asked how many inches $50.2\,\mathrm{cm}$ is, knowing that one inch is $2.4\,\mathrm{cm}$.

We’re going to need to think about multiplication in a new way to deal with these situations.

4.2.1 Proportion

Before we work out how to perform calculations like these, I want to explore situations like Question 4.2.1(a) in more detail. If you just want to know the procedure for multiplying measuring-numbers, you can skip ahead to Section 4.2.4: A Previously Solved Problem. A calculator can tell you what $1.21\times0.07$ is, for example, but it can’t tell you which calculation to do in the first place. You will need to understand proportion to know that.

When something travels at a constant speed (without slowing down or speeding up), there’s a special relationship between the distance it’s travelled and the time taken. It’s called ‘proportion’, and you could say ‘the distance is in proportion to the time’ or ‘the distance is proportional to the time’. That means that if I pick a unit of time, for example $1$ minute as in Question 4.2.1(a), then the distance traveled in that unit of time is always the same ($100\,\mathrm{m}$ in this case), no matter what part of the journey it’s in. The distance increases by $100\,\mathrm{m}$ for each extra minute.

If I changed my pace, walking more slowly for a while, then the distance traveled would not be in proportion to the time taken, because I might travel $100\,\mathrm{m}$ in the first minute, but then only $90\,\mathrm{m}$ in another minute.

Question 4.2.2

There are many different situations that involve proportion. Here are some more examples:

  1. If you have a map where every inch on the map represents $5$ miles in real life, and the distance between two towns is $5$ inches on the map, how far apart are the towns? The distance on the map is proportional to the distance in real life. (On a world map, where the surface of the globe has to be distorted to fit on a flat piece of paper, the distances on the map are not in proportion to the distances in real life.) Show answer

  2. A recipe requires $20$ grams of sugar for every cup of water. If you want to use $80$ grams of sugar, how much water should you use? In this case, the mass of the sugar is proportional to the volume of the water (and the volume is proportional to the mass – proportion goes both ways). Show answer

  3. If a pump can fill a $120$ litre tank in $15$ minutes, how much water can it pump in $1$ minute? The volume of water is proportional to the time. Show answer

Now that you have a feel for what proportion is, let’s see what happens when we’re not restricted to counting-numbers.

Question 4.2.3

Imagine you have a machine that turns seawater into clean drinking water. The machine works day and night, and a constant stream of water comes out – a total of $4$ tonnes every day (every $24$ hours). The water goes into a tank. At the start the tank is empty, or in other words, after $0$ days there is $0$ tonnes of water in the tank. After $1$ day there are $4$ tonnes, after $2$ days there are $8$ tonnes, after $3$ days there are $12$ tonnes, and so on. We can write down these facts as equations: \begin{align*} 4\times0 &= 0\\ 4\times1 &= 4\\ 4\times2 &= 8\\ 4\times3 &= 12 \end{align*}

  1. How much water comes out of the machine in half a day ($12$ hours)? How should we write this down as an equation? A ‘half’ is actually a number, and is usually written as $\frac{1}{2}$. Show hint

    Show answer

  2. How much water comes out in $2$ and a half days? How should we right this down as an equation? Two and a half is also a number, and is written as $2\frac{1}{2}$. Show answer

  3. How much water comes out in $0.1$ days? It might help you to consider how big the answer will be – is it greater or less than $1$? How should we write this down as an equation? Show hint

    Show answer

  4. Suppose that at midnight on Monday, you’ve forgotten how much water there is in the tank, so you mark the current level, and decide to talk about the amount of water in the tank as the number of tonnes more than the Monday midnight level. So $3$ days later the amount of water is $+12$ tonnes, which we could write as $4\times3 = 12$, or to be clearer, $4\times(+3) = +12$. How much water was there $2$ days before, at midnight on Saturday? How should we write this as an equation? Show hint

    Show answer

With the old kind of multiplication, we could only ask how much water would come out in $1$ day, $2$, $3$, $4$ days, and so on. Now we could ask how much comes out in an hour, a minute, a second, and even smaller units of time if we wanted. Of course, with any real machine, there would be small changes in the flow of water so that the mass of the water isn’t actually in proportion to the time when we use small units like seconds. In fact, all of the examples we’ve looked at aren’t really proportion if you look closely enough. For example, the grains of sugar prevent you from having exactly $20$ grams for every cup of water even if you had a scale that could measure that accurately, but it’s still very useful to treat these situations as if they were true mathematical proportion. In Section 4.2.1: Average Rate we’ll talk more about treating situations that aren’t proportional as if they were. For the rest of this section, we’re going to do some proportion problems requiring a bit more ingenuity.

Instead of saying ‘for every’, people often say ‘per’. For example, ‘the car was travelling at $20$ kilometres per hour’ means that it went $20$ kilometres for every hour.

Question 4.2.4

If a car travels at $8$ miles per hour, how far does it go in $15$ minutes? Show hint

Show answer

Question 4.2.5

A wheel turns around $500$ times every $4$ minutes, and travels $80$ centimetres per turn.

  1. What is the ‘rpm’ of the wheel? That stands for ‘revolutions per minute’ – the number of turns the wheel makes in $1$ minute. Show answer

  2. How far does it travel in $1$ minute? Show answer

  3. There is a mark somewhere on the wheel. If the mark is at the top of the wheel when it starts rolling, and it stops as soon as the mark gets to the bottom, how far has the wheel travelled? Show hint

    Show answer

Question 4.2.6
  1. There is a story about Archimedes that says a king asked him to work out if a crown was pure gold, or a mix of gold and silver. Archimedes found out by measuring the mass and volume of the crown. Gold weighs $20$ grams for every millilitre, and silver weighs $10$ grams for every millilitre. If the crown weighed $250$ grams, and its volume was $15$ millilitres, was it pure gold? Show answer

  2. Guess how many millilitres of gold and how many of silver went into the crown, and see if you are right. (This question is answered with algebra in Question 7.5.36.) Show hint

    Show answer

Question 4.2.7
  1. The road from Town A to Town B is $360$ kilometres long. A car travels along the road from town A towards town B at $40$ kilometres per hour. Another car travels from town B towards town A at $50$ kilometres per hour. If they both leave at the same time, how far from town A will they be when they meet? Show hint

    Show answer

  2. A hiker starts walking along a trail at $80$ metres per minute. Six minutes later, a runner starts running along the trail at $200$ metres per minute. When will the runner overtake the walker? Show hint

    Show answer

  3. A tank is being filled with water through two pipes. $80$ litres of water per minute flow through the first pipe. $600$ litres of water flow into the tank in $5$ minutes. How much water flows through the second pipe every minute? Show answer

  4. If pump A takes $30$ minutes to fill a $4500\,\mathrm{L}$ tank, and pump B takes $45$ minutes, how long will they take to fill the tank if they’re both pumping together?

Question 4.2.8

You want to fill an aquarium with salt water. Your fish need $110$ litres of water, with $35$ grams of salt for every litre (in other words, the ‘concentration’ of the salt is $35$ grams per litre). You already have $100$ litres of salt water, but with a salt concentration of only $30$ grams per litre, and you want to add some salt and water to it to get the salt water you need. How much salt do you need to add? Show hint

Show answer