Parallelogram 14 • Level 3 • 7 Dec 2023Chrismaths - Part 1

The direct method is to do the subtraction sum, as follows: 60,000 - 12 = 59,979.

Without doing the full subtraction we can see that the units digit of 60 000 − 21 is 9. In the context of the JMC you are entitled to assume that one of the given options is correct. As 59,979 is the only option with units digit 9, we deduce that this is the correct answer.

Since 8 km is approximately 5 miles, 1 km is approximately 58miles.

Hence 1.2 km is approximately 1.2×58 miles.

We have 1.2×58=1.2×58=68=0.75.

Therefore 1.2 km is approximately 0.75 miles.

Alternatively, since 8 km is approximately 5 miles, 1 mile is approximately 85 km, that is, 1.6 km.

It follows that 1.2 km is significantly less than 1 mile. So 0.75 miles is the correct option.

More precisely, it follows that 1.2 km is approximately 1.21.6 miles. Since:

1.21.6=1216=34=0.75,

it follows that 1.2 km is approximately 0.75 miles.

The song refers to 525,600 minutes. Since there are 60 minutes in an hour, 525,600 minutes is:

52560060 hours, that is, 8760 hours.

Since there are 24 hours in one day, 8760 hours is:

876024 days, that is, 365 days.

It follows that of the given options a year is closest to the length of time corresponding to the number of seconds mentioned in the song.

Note A calendar year has 365 days in a normal year, but 366 days in a leap year such as 2016.

Astronomers use several different definitions of a year which differ a little from each other in length. One common definition is the sidereal year which is the time it takes the earth to complete one revolution about the sun, relative to the distant stars. The sidereal year has length 365.256363004 days.

Because this is not an exact number of days, a calendar with the same number of days in each year would not be synchronised with the position of the earth relative to the sun and so it would get out of step with the seasons.

Different cultures have solved this problem in different ways. In the most commonly used calendar, introduced by Pope Gregory XIII in 1582, the rules for the length of a year are:

A year is a leap year if its number is a multiple of 4, except that if the number is a multiple of 100 but not of 400, then it is not a leap year. A standard year has 365 days. A leap year has 366 days.

For example, 2016 is a multiple of 4. Therefore the current year is a leap year. 2017 is not a multiple of 4, so next year will not be a leap year. 2000 is a multiple of 100 and also of 400. So the year 2000 was a leap year. However, 2100 is divisible by 100 by not by 400. So the year 2100 will not be a leap year.

Since the ratio of boys to girls is 3 : 5, the proportion of boys in the group is 38. Therefore the number of boys is:

38×48=18.

It follows that the number of girls in the group is 48−18=30.

The ratio 5 : 3 is the same as the ratio 50 : 30. So in a group which has 30 girls, in which the ratio of boys to girls is 5 : 3, there will be 50 boys. The number of boys is 18. To make this up to 50, the number of boys that need to join the group is 50−18=32.

The four triangles between them have 12 edges. When they are fitted together there must be at least 3 places where two of the triangles meet along an edge. At each of these places 2 of the 12 edges form interior edges of the parallelogram. So the perimeter of the parallelogram will be made up of at most 12−3×2=6 of the edges of the triangles. So, if we can arrange the triangles so that the perimeter is made up of all four edges of length 13 cm, and two edges of length 12 cm, this would give us the largest possible perimeter.

The diagram shows that such an arrangement is possible. So the length of the longest possible perimeter is 4 × 13 cm + 2 × 12 cm = 76 cm.