AMO 2027 Grades 6-7 SET 10 - Self-study

Alice and Danny cycled from Town X to Town Y at 12 km/h and 10km/h respectively. Alice left Town X at 9.00 a.m and arrived at Town Y at 9.30 a.m When Alice arrived at Town Y, Danny was 1.5 km away from Town Y.

a) What is the distance between the 2 towns?

b) What time did Danny leave Town X?

**Editorial input note:** Enter the distance in kilometres, then the departure time on a 24-hour clock using HH:MM; omit units. Separate the two parts with a semicolon followed by one space; use no space before the semicolon.

There are 60 girls and 75 boys who joined the AEP Math Trail. How many groups that consist of same number of girls and boys can be formed? How many girls and how many boys will be in each group?

Find the LCM of

a) 9, 16, 24.

b) 8, 36, 60.

**Editorial input note:** Enter the answers to (a) and (b), in order, separated by a semicolon with no spaces.

1/3 of the pupils in a hall were boys. When 35 boys left the auditorium, the number of boys decreased to 1/4 of the total number of pupils in the hall. How many girls were in the hall?

15 identical rectangles of perimeter 20cm each are arranged as shown below. Find the perimeter of the figure.

The cost of a T-shirt and a skirt was $\$85$. Janet bought 6 such T-shirts, 2 such skirts and a hat for $\$430$. If the cost of 1 T-shirt and 1 hat was $\$125$, find the cost of 1 skirt.

A number has a remainder of 2 when it is divided by 3. It has a remainder of 3 when it is divided by 4. Find the sum of two such numbers that are smaller than 30?

Find the missing number.

a) Three central squares with surrounding numbers (top, left, right, bottom; centre): (12, 18, 3, 12; 45), (19, ?, 13, 12; 55), (4, 8, 9, 3; 24).

b) 4 × 4 grid rows: [88, 35, 10, 1]; [35, 20, 7, 2]; [10, ?, 4, 1]; [1, 2, 1, 2].

c) Three 2 × 2 grids, rows top to bottom: [24, 11; 89, 76], [8, 6; 94, 92], [10, 15; ?, 90].

Complete matching diagram from the supplied Grades 8–9 duplicate, physical PDF page 33; recovery details are in the solution.



**Editorial input note:** Enter the missing numbers for (a), (b), and (c), in order, separated by semicolons with no spaces.

The ratio of the perimeter of an equilateral triangle to the perimeter of a square to the perimeter of a rectangle is 2 : 3 : 5. If the length and width of the rectangle are 36cm and 24cm respectively, how much longer is a side of the square than a side of the equilateral triangle?

Tom, Patty and Bart scored 306.5 marks in an English examination. Patty scored 18.5 marks more than Tom. Bart’s score was thrice that of Patty’s. If the passing mark was 100, how many more marks did Tom need to pass?

Triangle ABC is a right triangle with AB = 60 units, AC = 80 units and BC = 100 units. D is a point between B and C such that △ADB and △ADC have equal perimeters. Find the length of the side BD.

A. 10 units
B. 20 units
C. 40 units
D. 60 units
E. None of the above

Let the operation ∗ be defined by

$$a\ast b=ab-a-b+2.$$

If 7 ∗ b = 13, what is the value of b?

Rectangular blocks with a square base of sides 5 cm are packed into a box in layers as shown in the figure below.
If a maximum of 48 rectangular blocks are able to fit into the box, what is the length of the box?

In the diagram, M is the midpoint of YZ, the measure of ∠XMZ = 30°, and the measure of ∠XYZ = 15°. What is the measure of ∠XZY?

Thomas has $\$9$ more than Tom. Alice has $\$20$ fewer than Tom. The total amount of money they have is $\$199$. How much money does Thomas have?