In basic arithmetic there are four different operations
+ – × ÷
There is a type of question which turns up in various forms, all basically the same, which involve divide and multiply (division and multiplication if you prefer). I call this scaling.
Note there is a pdf version of this post created with latex and tikz which I recommend you read as
- latex and tikz enables the maths and graphs to be formatted better
- there are more example questions
On with post.
It makes sense to learn to recognise this type of problem and how to solve it as there are often several questions of this type in an 11+ or KS2 maths tests. Part of the art of succeeding at tests and exams is to be able to answer as many questions as you can as quickly as possible, in order to
a) secure as many marks as possible as early as possible
A key point in answering questions in exams and tests is to answer the easy question first. You do not (usually) have to answer the questions in the order they are set.
b) leave more time for harder questions
So if you know there is a type of question which may crop up more than once in various different forms it makes sense to learn how to recognise and solve these problems quickly.
Here is a simple example
If 2 ice creams cost £2.40 how much would 3 cost?
To solve this you have to first find out how much 1 ice cream costs
Divide the £2.40 (the price of 2 ice creams) by 2
So 1 ice cream costs £1.20
Then multiply by £1.20 by 3 to get £3.60 for the price of 3 ice creams.
I usually start explaining how to solve these problems like this, doing the divide first and then the multiply to stress why these steps are done, i.e. to find the price of one and once you know this you can multiply to find the price of any number.
If you really want to stress the steps it’s as well to say these questions
- only involve divide and multiply
- never add and subtract
Later you can explain for sums involving only multiplication and division they can be worked out in any order, so each of the following all have the same value
Note the sums in brackets are worked out first
and it make sense to choose the one that’s easiest to work out. Here there’s not much in it, but if the numbers we had to multiply and divide by where larger, for example
If 22 ice creams cost £2.40 how much would 33 cost?
It makes sense to see the problem as
And then it’s probably easiest to switch to
In KS2 the numbers are usually easy, in 11+ the numbers can involve 3 decimal places.
There is a variation of this type of problem where you start with the price of 1, for example
Butter costs 1.56 a kilogram how much would 1.6 kg cost?
There are 900 words in each chapter of a book. In 12 chapters how many words are there?
Sometimes children get confused as they don’t have to do the divide.
In 11+ maths exam often starts with questions that just tell you to calculate something for example
Calculate 1.2 × 1.43
Later on the questions are usually asked in words which mean whoever is taking the test has to translate the words into a series of arithmetic calculations.
Here are some example questions (there are more in the pdf version of this post) which show some of the different ways this type of scaling problem can come up, occasionally there are variations with also include + and -.
1/7 of a number is 9.
What is 1/3?
Butter costs £1.47 a kilogram how much does 1.92 kilograms cost.
The pie chart gives the percentages of different types of fruit in pupils lunch boxes at a school.
If there are 11 lunch boxes with apples how many lunch boxes have bananas?