Winchester College — 13+ Maths Entrance Examination 2022 (Interactive)

Q1(a): Find 37 of 42.

Q1(b): Find 30% of 150.

Q2(a): 24, written as the product of its prime factors, is 2^3 x 3. Write 18 as the product of its prime factors.

Q2(b): What is the highest common factor of 18 and 24?

Q2(c): 60 = 2^2 x 3 x 5 and 75 = 3 x 5^2. What is the lowest common multiple of 60 and 75?

Q3(a): Calculate 11 x 121.

Q3(b): Calculate 5432 - 433.

Q3(c): Calculate 5432 - (5432 - 19).

Q3(d): Calculate (sqrt(169))^2.

Q3(e): Calculate (133 + 134 + 135) / 3.

Q3(f): Calculate 2 x 678 x 5 + 9.

Q3(g): Calculate sqrt(640000).

Q3(h): Calculate 6 / 0.4.

Q4(a): Find in the simplest form: 1 113 - 413.

Q4(b): Find in the simplest form: (397) x (1433) x (1113).

Q4(c): Find in the simplest form: (917) / (4551).

Q4(d): Find in the simplest form: (12 + 14) / (13 + 16).

Q5(a): Find 48% of 50% of 400.

Q5(b): Evaluate the cube root of (9 x 33 x 121).

Q5(c): Evaluate sqrt(2748).

Q5(d): Evaluate (-2)^17 / ((-2)^4 x (-2)^8).

Q6(a): Solve 3x + 7 = 40.

Q6(b): Solve 56 / (2x + 1) = -8.

Q6(c): Solve sqrt((3 + 2x/5)^3) = 8.

Q6(d): a and b are integers. b is 80% more than a. Find and simplify the ratio a : b.

Q7(a): Given a = 5, b = -3 and c = 17, find the value of 14a / (b + c).

Q7(b): Given a = 5, b = -3 and c = 17, find the value of a^3 / 25.

Q7(c): Given a = 5, b = -3 and c = 17, find the value of (100a + 100b + 100c) / (a + b + c).

Q7(d): Given a = 5, b = -3 and c = 17, find the value of sqrt(2a - 3b + 6c).

Q7(e): Given a = 5, b = -3 and c = 17, find the value of (c^2 - a^2) / (c - a).

Q7(f): Given a = 5, b = -3 and c = 17, find the value of (a^2 + ab + b^2)(a - b).

Q8(a): In the diagram, a triangle has angle 36 degrees at the top-left vertex. Three lengths in the triangle are marked as equal (with tick marks), creating two sub-triangles. Find the angle a at the right vertex. (Refer to original PDF for figure.)

Q8(b): The diagram shows four lines, two of which are parallel (marked with arrows). Angles of 29 degrees and 97 degrees are shown at intersections. Find the angle b at the lower-left intersection. (Refer to original PDF for figure.)

Q9(a): Order from smallest to largest: 4sqrt(3), sqrt(50), 7, 3sqrt(5), 2sqrt(13). (Hint: try squaring the numbers.)

Q9(b): Order from smallest to largest: 295 x 305, 290 x 310, 299 x 301.

Q9(c): Order from smallest to largest: 12, 49, 613, 1021, 511, 817.

Q10(a): Each of the small circles has radius 2. The innermost circle is just touching the six which surround it, and each of those circles are just touching each of their neighbours and the large circle. Determine the circumference of the large circle. Leave pi in your answer. (Refer to original PDF for figure.)

Q10(b): Each of the small circles has radius 2. The innermost circle is just touching the six which surround it, and each of those circles are just touching each of their neighbours and the large circle. Determine the shaded area. Leave pi in your answer. (Refer to original PDF for figure.)

Q11(a): Find the median of the square numbers from 1 to 169 (inclusive).

Q11(b): Four friends are aged 12, 12, 15 and 17. They are by themselves in a room. Four more people, all the same age, enter the room and the mean age increases by 3. How old are the newcomers?

Q11(c): Three sisters are aged 8, 9 and 14. When their friend Kelly is with them the mean of their ages is equal to the median. How old could Kelly be? (There is more than one answer.) Give all possible answers.

Q12(a): Harry is training for a running race. A training session consists of a 1 minute sprint followed by a 30 second jog repeated 10 times. He sprints twice as fast as he jogs. How long is the entire training session?

Q12(b): In the training session Harry travels exactly 6 km. For what fraction of the distance was he sprinting?

Q12(c): Exactly how long has Harry been travelling for when he has travelled 2 km?

Q13(a): Chloe writes a list, in order, of all whole numbers that are written using no digits other than 1s and 5s. The list begins: 1, 5, 11, 15, 51, ... How many numbers in the list are less than 1000?

Q13(b): Chloe writes a list, in order, of all whole numbers that use no digits other than 1s and 5s. Which number in this list comes directly before 115511?

Q13(c): Chloe writes a list, in order, of all whole numbers that use no digits other than 1s and 5s. What is the 32nd number in the list?

Q13(d): Chloe writes a list, in order, of all whole numbers that use no digits other than 1s and 5s. What is the average of all the numbers in the list that are more than 100,000 but less than 1,000,000?

Q14(a): The diagram shows two right-angled triangles. One has a right angle at the bottom-right with base 10 and vertical side 5. The other has a right angle at the top, with a vertical side of 2. The two triangles share a common hypotenuse. Find the value of x, where x is a side of the upper triangle. (Refer to original PDF for figure.)

Q14(b): The diagram shows two right-angled triangles. One has a right angle at the bottom-right with base 19 and vertical side 12 (split into 4 at top and 8 at bottom). The sloped line y connects two vertices. Find the value of y. (Refer to original PDF for figure.)

Q14(c): ABCDEFGH is a box in the shape of a cuboid. A thin rod of length 21 cm just fits inside the box with one end at A and the other at G (the space diagonal). If the same rod has one end at A and passes through C, exactly 1 cm sticks out of the box. If the rod passes through H, exactly 2 cm sticks out. If the rod passes through F, exactly z cm sticks out. Find z. (Refer to original PDF for figure.)