Dear Uncle Colin,

I have a question I can’t make sense of: “It takes 3 bricklayers four hours to lay 4200 bricks. How long would it take 2 bricklayers to lay 3150 bricks?” I can never figure out when to multiply and when to divide!

Markscheme Obtusely Recommends Taking A Ratio

Hi, MORTAR, and thank you for your message!

My key trick to working out when to divide and when to multiply is to ask: *would changing this number make the workers lay more bricks or fewer?"

### Thinking it through (1)

For instance, with this one, we know that three bricklayers lay 4200 bricks in four hours. What happens if we only have two bricklayers, but keep the time the same? We lay fewer bricks, two-thirds as many. That’s 2800 bricks.

But we want 3150 bricks. That will take the two builders longer - by a factor of $\frac{3150}{2800} = \frac{9}{8}$. Nine-eights of four hours is four and a half hours, which is the answer.

### Thinking it through (2)

Depending on how happy you are with the fractions, that might be a bit much, so here’s another method: take things down to *ones*. One bricklayer, one hour. Then see where you are.

If 3 bricklayers take four hours to lay 4200 bricks, one bricklayer takes four hours to lay 1400. They would take one hour to lay 350 bricks.

That means, two bricklayers lay 700 bricks in an hour.

How many hours do they take to lay 3150? Well, it’s certainly more than one - if you’ve got your estimating hat on, it’s between 4 and 5. If you then work out 3150 divided by 700, it give you four and a half hours again.

I hope that helps!

- Uncle Colin

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

## Barney M-T

Thanks C for highlighting this sort of question – it certainly causes much trouble at GCSE, and I don’t mind admitting I find it difficult to teach. My daughter (Year 11) had this topic set for homework the other day, and unusually there was no supporting video on her school’s system (probably because it’s so hard to teach).

I’d argue there isn’t even an accepted name for this type of question – making it hard to look up in the text books! Pinpoint Maths calls it “changing work rate problems”.

My favourite method is as follows:

FOR 4200 BRICKS: 3 workers x 4 hours = 12 man hours [measuring the “size of the the job”]

FOR 3150 BRICKS: 12/4200×3150 = 9 man hours [measuring the size of the new job. The logic here is as for Colin’s second method: how big would the job be for 1 brick (divide by 4200) then for 3150 bricks (x3150). Very similar to currency conversion problems].

so 4.5 hours for each worker.

Two issues with my method though:

1) most teens are not familiar with the concept of “worker hours” as a measure of how big the whole job is. But I think it’s a useful concept for them to grasp.

2) the language sometimes has to be adapted e.g. “worker days”.

Do others think it is acceptable to use “man hours” instead of the more inclusive “worker hours” – purely in the interests of less writing?

## Colin

“I would try not to use language that excludes about half the population, but it’s THREE MORE LETTERS!” I know, as mathematicians, we’re meant to be lazy, but come on.

## Rebecca Gilardoni

I have my maths exam in 5 days time and I come across this website.

It’s really helpful Colin thank you.

Becky

## Colin

Good luck, Becky!