Ratios

a) In order to work out the fraction of students that have blonde hair, we need to add up the ratio parts. The sum of the ratio is 4 + 5 = 9.  This means that we are dealing with 9ths. Since the ratio share for blond students is 4, this means that the fraction of blonde students is \dfrac{4}{9}

 

b) We know from the previous question that \dfrac{4}{9} of the students have blonde hair.  Therefore, the fraction of students with brown hair is \dfrac{5}{9}.  If there is a total of 450 students in the school, we need to work out what \dfrac{5}{9} of 450 is:

 

\dfrac{5}{9} \times 450 = 250 students.

In total, there are 7 parts to this ratio (2 + 5 = 7).  If 7 shares have a value of 35, then 1 share has a value of 5 (35 \div 7 = 5).

 

Since 1 share has a value of 5, then 2 shares will have a value of 10 (2 \times 5 = 10).

 

Since 1 share has a value of 5, then 5 shares will have a value of 25 (5 \times 5 =  25).

To work out the total cost of the tiles Lucy buys, we need to work out how many white tiles she buys. In order to work out the number of white tiles,  we  need to work out the total number of tiles she buys.

 

The ratio is 2 parts blue to 13 parts white.  If Lucy buys 16 blue tiles, this is 8 times more than the figure for blue tiles in the ratio (16\div2=8).

If the number of blue tiles she buys is 8 times more than the blue tiles figure given in the ratio, then the number of white tiles she buys must also be 8 times more than the white tiles figure in the ratio.  Therefore, the number of white tiles she buys is:

13\times8 = 104 white tiles

 

Now that we know how many tiles of each colour she has bought, we can calculate the total cost of  the tiles.

The cost of 16 blue tiles is:

16\times2.80 = \pounds44.80

The cost of 104 white tiles is:

104\times2.35 =\pounds244.40

 

Therefore, Lucy’s total spend on tiles is:

44.80+244.40=\pounds289.20

The first thing we need to do is to deduct the 20 \% spent on the magazine subscription so that we can work out how much of his allowance Steve has left over.  20 \% of £200 can be calculated as follows:

0.2 \times \pounds 200 = \pounds 40

You may prefer to calculate the 20% in your head:

10 \% of £200 is £20, so 20 \% is

2 \times \pounds 20 = \pounds 40

 

Then deduct this from £200:

\pounds200 - \pounds40 = \pounds160

 

Steve therefore has £160 pounds remaining which he spends on sweets, football stickers and fizzy drinks in the ratio of 5 : 2 : 1.

By adding up the ratio, we know that we are dealing with eighths.  (We know we are dealing in eighths because 5 + 2 + 1 = 8).  We know from the ratio that the share he spends on football stickers is 5, meaning that Steve spends \frac{5}{8} of the remaining allowance on football stickers.

 

The actual amount that Steve spends on football stickers can be calculated as follows:

\dfrac{5}{8} \times \pounds160 = \pounds100

a) We are told that Jon reads twice as many books as Kate. As a ratio, this can be written as 2 : 1.

We are also told that Alieke reads 4 times as many books as Jon.  As a ratio, this can be written as 4 : 1.

 

The issue we have now is that in the Jon : Kate ratio, Jon’s share is 2, while in the Alieke : Jon ratio, Jon’s share is 1.  By doubling the 4 : 1 ratio for Aleike : Jon to 8 : 2, Jon’s share is now the same in both ratios.

This means that we can express this is a 3-way ratio as follows:

\text{Alieke : Jon : Kate }= 8 : 2 : 1

 

b) We are told that Alieke read 63 more books than Kate last year, and we know from the previous part of the question that the Alieke : Kate reading ratio is 8 : 1. In this ratio, the share is 8 parts to 1 part, so we can conclude that the difference between the ratio share is 7 parts (8 - 1 = 7).

 

If the difference in the ratio share is 7 parts, and the difference in the number of books read is 63, then we can work out the number of books read that 1 share of the ratio represents:

63 \div 7 = 9\text{ books}

 

If one share of the ratio represents 7 books read, we can now work out how many books were read in total by the three people.  By adding up the ratio, we know that the total number of shares is 11 (8 + 2 + 1 = 11), so the total number of books read can be calculated as follows:

11 \times 9\text{ books} = 99\text{ books}