Plenty of pens
Amy's mum had given her £2.50 to spend. She bought four times as many pens as pencils and was given 40p change. How many of each did she buy?
Problem
Amy went into her local stationery shop. Her mum had given her £2.50 to spend.
Amy liked the look of some luminous pens, which cost 15p each, and some fancy pencils, which cost 10p each.
She bought four times as many pens as pencils and was given 40p change.
How many of each did she buy?
Getting Started
How much money did Amy actually spend?
Can you use this amount to estimate how many pens and pencils she might have bought?
Student Solutions
Adrian has worked out this problem very efficiently:
No of pencils $= x$
No of pens $= 4x$Spent on pencils $= 10x$
Spent on pens $= 4x \times15 = 60x$
Total spent $= 10x + 60x = 70x$$70x = 250 - 40$
$70x = 210$
$x = 210 / 70 = 3$Therefore no of pencils $= 3$
No of pens $= 3 \times4 = 12$
Pupils at Moorfield Juniors, including Emma, approached it slightly differently:
We worked this out by taking 40p from £2.50 which is 2.10 then using trial and error.
It was easy because there aren't many low numbers in the 4 times table.
The answer is 12 pens which equals £1.80. 12 divided by 4 = 3 so there are 3 pencils which equals 30p.
£1.80+30p= £2.10.
Teachers' Resources
Plenty of Pens
Amy went into her local stationery shop. Her mum had given her £2.50 to spend.
Amy liked the look of some luminous pens, which cost 15p each, and some fancy pencils, which cost 10p each.
She bought four times as many pens as pencils and was given 40p change.
How many of each did she buy?
Why do this problem?
This problem can be approached in several different ways (by randomly trying numbers, working in a systematic way to try numbers, making a table ...) and could be a nice way of introducing algebra.
Possible approach
There are two constraints in this question - the relationship between the number of items bought and the total price.
As an introduction you might wish to start with a similar problem and focus on just one constraint. Choose two similar objects - perhaps a dice and a ruler, and allocate them different prices, say 20p and 30p. Ask the children how many objects you could buy if you had, say, £2.00 to spend.
After they have worked for a while, bring the class together and record their results on the board. (You might do this in a table, or randomly record them and ask the children whether there mght be a better way.)
Ask some questions - what's the greatest number of items I could have bought? The smallest?
Then offer the children the task as written and some time to explore.
Bring them back together again and share their results. If appropriate, you could compare approaches, or together take a look at the submitted solutions and discuss the merits of each.
Key questions
Possible extension
Learners could try Buckets of Thinking which involves similar mathematics.
Possible support
Using real pencils and pens may help children keep track of what they have tried.