Video: AQA GCSE Mathematics Higher Tier Pack 2 β€’ Paper 1 β€’ Question 26

In a school, you can wear summer shirts or normal shirts. In year seven, the ratio of boys to girls is two to three. 25 percent of the girls are wearing summer shirts. 40 percent of the boys are wearing summer shirts. 62 children are wearing summer shirts. Work out the total number of children in year seven.

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Video Transcript

In a school, you can wear summer shirts or normal shirts. In year seven, the ratio of boys to girls is two to three. 25 percent of the girls are wearing summer shirts. 40 percent of the boys are wearing summer shirts. 62 children are wearing summer shirts. Work out the total number of children in year seven.

A question like this can be quite intimidating as there is so much information. We will look at two different methods of working out the final answer: the total number of children in year seven. For our first method, we will let 𝑑 be the total number of children. This is the answer we want to work out. The first bit of information tells us that the ratio of boys to girls is two to three. Adding two and three gives us five. Therefore, we can split the total number of children 𝑑 into five parts. As the boys corresponded to two parts, we can see that the number of boys is two-fifths of 𝑑. As the number of girls was three parts of the ratio, the number of girls is three-fifths of 𝑑.

Our second bit of information tells us that 25 percent of the girls are wearing summer shirts. 25 percent is equal to one-quarter. Therefore, one-quarter of the girls are wearing summer shirts. We can therefore work out an expression for the number of girls wearing summer shirts by finding a quarter of three- fifths 𝑑. In maths, the word of means β€œmultiply”. Therefore, we need to multiply one- quarter by three-fifths 𝑑. To multiply two fractions, we multiply the numerators and then separately multiply the denominators. One multiplied by three is equal to three, and four multiplied by five is equal to 20. Therefore, the number of girls wearing summer shirts is three-twentieths of 𝑑.

The number of boys wearing summer shirts was 40 percent. And we know that 40 percent is equal to four-tenths as a fraction. This fraction can be simplified by dividing the numerator and denominator by two. Remember whatever you did the top, you must do to the bottom. Four divided by two is equal to two, and 10 divided by two is equal to five. Therefore 40 percent, is equal to two-fifths. The number of boys wearing summer shirts is therefore equal to two-fifths of two-fifths 𝑑. Once again, we can calculate this by multiplying two-fifths by two-fifths 𝑑. Two multiplied by two is equal to four, and five multiplied by five equals 25. Therefore, the number of boys wearing summer shirts is four twenty-fifths of 𝑑. Finally, we were told that 62 children are wearing summer shirts. This means that the number of boys wearing summer shirts and the number of girls wearing summer shirts must add up to 62.

This can be written as an equation four twenty-fifths of 𝑑 plus three-twentieths of 𝑑 equals 62. In order to add two fractions, we must ensure that we have a common denominator, the denominator needs to be the same. The lowest common denominator of 25 and 20 is 100 as 100 is the lowest number in the 25 and 20 times tables. Four twenty-fifths is equal to sixteen hundredths as four multiplied by four is equal to 16 and 25 multiplied by four equals 100. In the same way, three-twentieths is equivalent or the same as fifteen one hundredths as three multiplied by five is equal to 15 and 20 multiplied by five equals 100. We can therefore rewrite the equation as sixteen one hundredths of 𝑑 plus fifteen one hundredths of 𝑑 is equal to 62. Adding these two fractions gives us thirty-one one hundredths of 𝑑 as 16 plus 15 is equal to 31.

We now need to solve this equation to work out our value of 𝑑. Firstly, we can multiply both sides of the equation by 100. This gives us 31𝑑 is equal to 62 multiplied by 100. Dividing both sides of this equation by 31 gives us 62 multiplied by 100 divided by 31. As 31 is a half of 62, this can be cancelled leaving us with 𝑑 is equal to two multiplied by 100. Two multiplied by 100 is equal to 200. As 𝑑 was the total number of children in year seven, we can say that there are 200 students in year seven. An alternative method would be to set up a two-way table. We have the number of boys, the number of girls, and the total. We also have the number of summer shirts, the number of normal shirts, and the total number of shirts. Our aim is to work out the total number of children altogether.

Firstly, we will let 𝑏 be the number of boys and 𝑔 be the number of girls. This means that the total number of pupils is 𝑏 plus 𝑔. There were 62 children in total wearing summer shirts. 40 percent of the boys were wearing summer shirts, and we know that 40 percent is equal to two-fifths. Therefore, the number of boys wearing summer shirts is two-fifths of 𝑏. 25 percent of the girls are wearing summer shirts, and 25 percent equals a quarter. Therefore, the number of girls wearing summer shirts is one-quarter of 𝑔. Two-fifths of 𝑏 plus one quarter of 𝑔 is equal to 62. We also know that the ratio of boys to girls is two to three. This can be rewritten as a fraction, so the number of boys divided by the number of girls is equal to two-thirds. Multiplying both sides of this equation by 𝑔 gives us 𝑏 is equal to two- thirds of 𝑔. This means that the number of boys is equal to two-thirds of the number of girls.

We can now substitute 𝑏 equals two-thirds 𝑔 everywhere we see a 𝑏 in the table. The total number of boys is two-thirds 𝑔. The number of boys wearing summer shirts is two- fifths multiplied by two-thirds 𝑔. This is equal to four fifteenths 𝑔. Two multiplied by two is equal to four, and five multiplied by three equals 15. Finally, the total number of students altogether is two-thirds 𝑔 plus 𝑔, the total number of boys plus the total number of girls. We can now write the top line of the table, the number of students with summer shirts, as an equation: four fifteenths 𝑔 plus one-quarter 𝑔 is equal to 62. Once again, we need to find a common denominator to add the two fractions. The lowest common multiple of four and 15 is 60. Multiplying the numerator and denominator of the first fraction by four gives us 16 over 60 𝑔. Multiplying the numerator and denominator of the second fraction by 15 gives us 15 over 60 𝑔.

We are now left with sixteen sixtieths 𝑔 plus fifteen sixtieths 𝑔 is equal to 62. Adding the two numerators gives us 31; 16 plus 15 is equal to 31. So we have thirty-one sixtieths of 𝑔 is equal to 62. We can then multiply both sides of this equation by 60. This gives us 31 𝑔 is equal to 62 multiplied by 60. Dividing both sides of the new equation by 31 gives us 𝑔 is equal to 62 multiplied by 60 divided by 31. Once again, we can cancel the 62 and 31. So 𝑔 is equal to two multiplied by 60. Two multiplied by 60 is equal to 120. Therefore, there are 120 girls altogether in year seven. As there are 120 girls, we can work out the number of boys by finding two-thirds of this number. One-third of 120 is 40 as 120 divided by three equals 40. This means that two-thirds of 120 is equal to 80. There are 80 boys altogether in year seven. As there are 80 boys and 120 girls, we have once again proved that there are 200 children altogether in year seven.

At this stage, we could also complete the rest of the table. 25 percent of the girls were wearing summer shirts, and 25 percent of 120 is 30. 40 percent of the boys are wearing summer shirts, and 40 percent of 80 is 31. 32 plus 30 is equal to 62. We could use this information to work out the number of boys, girls, and total students wearing normal shirts. 80 minus 32 is equal to 48; 120 minus 30 is equal to 90; and 200 minus 62 is equal to 138. We now have a complete table showing the types of shirts that boys and girls were wearing.

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