Bananas come in bunches of 3 (small bunch) or 5 (large bunch). Tim wants to purchase exactly 32 bananas; how many different combinations of bunches could he purchase?

*Solution*: The possible bunch combinations are 4 small and 4 large, or 9 small and 1 large, so there are just **two **combinations with a total of 32 bananas

How many three-digit numbers contain the digit 3?

*Solution*: There are 9x10x10=900 three-digit numbers, of which 8x9x9=648 do not contain the digit 3, so the required total is 900-648=**252**

ive angles in a hexagon are 110, 120, 130, 140 and 150 degrees. What is the sixth angle?

*Solution*: The angles in a hexagon sum to 720, so the sixth angle is 720-(110+120+130+140+150)=720-650=**70 degrees**

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 4 is 1/4, and the reciprocal of 2/5 is 5/2. Find all numbers that are quadruple their reciprocal.

*Solution*: Let x be such a number. Then x = 4/x, so x^2 = 4, so x=2 or -2. So there are two such numbers: **2 **and **-2**.

What is the sum of all the odd numbers between -1000 and 990?

*Solution*: Most of the numbers in the sum cancel each other out; this happens with the numbers between -990 and 990. So the required sum is (-999)+(-997)+(-995)+(-993)+(-991) = **-4975**

Find three positive integers less than 1000, each of which is both a square and a cube.

*Solution*: There are only three such integers; all of them are sixth powers. 1x1x1x1x1x1=1, 2x2x2x2x2x2=64 and 3x3x3x3x3x3=729. **1, 64, 729**

rectangle has a length 4 more than its width, and a perimeter of 28. What is its area?

**Solution:** The dimensions are 9 and 5, so the area is **45**

A rectangle has a perimeter of 21 and an area of 5. What is its length?

**Solution:** The dimensions are ½ and 10, so its length is **10**

If x is 2/3 of y, and z is 4/5 of x, what fraction of z is y?

**Solution**: z=(4/5)x=(4/5)(2/3)y=(8/15)y. So y=(15/8)z. That is, y is **15/8** of z.

What is the largest three-digit multiple of 6 that is still a multiple of 6 when its digits are reversed?

**Solution**: A number is divisible by 6 precisely when it is even and its digits add to a multiple of 3. So we require the first and last digits to be even. The largest number satisfying the given conditions is **894**.

Find three positive integers, each of which has an equal number of even factors and odd factors.

**Solution**: Any number divisible by 2 but not by 4 will do. The first few examples are 2 (factors are 1,2), 6 (factors are 1, 2, 3, 6) and 10 (factors are 1, 2, 5, 10)

Find three positive integers, each of which has three times as many even factors as odd factors.

**Solution**: Any number divisible by 8 but not by 16 will do. The first few examples are 8 (factors are 1, 2, 4, 8), 24 (factors are 1, 2, 3, 4, 6, 8, 12, 24) and 40 (factors are 1, 2, 4, 5, 8, 10, 20, 40)

The reciprocal of a number is the result of dividing 1 by the number. For example, the reciprocal of 4 is 1/4. What is the reciprocal of 1/2 + 1/3?

**Solution**: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, and the reciprocal of 5/6 is **6/5**

A positive integer (whole number) n is divisible by 14 and 15. How many other positive integers are guaranteed to be factors of n?

**Solution**: Since n is divisible by 14 and 15, it must be divisible by the lowest common multiple of 14 and 15, which is 210. It must also be divisible by all the factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210. So there are 16 factors of 210, of which we have already counted the factors 14 and 15; hence there are **14 **other positive integers guaranteed to be factors of n

Which negative number is four times as far away from 41 as from 2??

**Solution:-11 **(it differs from 2 by 13 and from 41 by 52)

What is the smallest cube divisible by 6 and 10?

**Solution**: The cube must be divisible by 2, 3 and 5. Further, in a cube, the exact power to which any prime appears in its prime factorisation must be a multiple of 3. So the smallest cube divisible by 6 and 10 is 2 cubed x 3 cubed x 5 cubed = **27000**

Four football teams play each other exactly once, with 3 points for a win, 1 for a draw and 0 for a loss. The teams are then ranked according to the number of points. Teams with the same number of points are awarded the same rank.

1) What is the smallest possible total number of points for the highest ranked team(s)?

2) What is the greatest possible total number of points for the lowest ranked team(s)?

*There are 6 games in total, and in each game, either 2 or 3 points are awarded overall. So the total number of points awarded is between 12 and 18, and the average number of points per team is between 3 and 4.5*

1) **Solution**: The highest ranked team cannot have fewer points than the average number of points per team, so they must score at least 3 points. This can happen is every match is a draw. So the smallest possible number of points for the highest ranked team is **3**

2) **Solution**: The lowest ranked team cannot have more points than the average number of points per team, so they cannot score more than 4 points. This can happen is every team wins one, draws one and loses one of their matches. So the greatest possible number of points for the lowest ranked team is **4**

What is the mean of 1, 10 and 100?

**Solution**: The mean is (1+10+100)/3=111/3=**37**

The mean of 2, n and 222 is 22. What is the value of n?

**Solution**: The mean of the three numbers is 22, so their sum is 66. Now we have 2+n+222=66, so n=66-222-2=**-158**

Write 27 as the sum of a positive multiple of 4 and a positive multiple of 5.

**Solution**: 27=**12+15** (12=3×4 and 15=3×5

In how many ways can 29 be written as the sum of a positive multiple of 6 and a positive multiple of 7?

**Solution**: The multiple of 7 could be 7, 14, 21 or 28. But we have 29=7+22=14+15=21+8=28+1, and none of 22, 15, 8 and 1 is a multiple of 6. So it is impossible to write 29 as required, and the answer is **zero**

How many three-digit numbers are divisible by 10 and 15?

**Solution**: An integer is divisible by 10 and 15 precisely when it is divisible by their lowest common multiple, which is 30. The smallest and largest three-digit multiples of 30 are 30×4=120 and 30×33=990 respectively. So the required total is 33-3=**30**

How many two-digit numbers are divisible by exactly one of the numbers 10, 15, 20?

**Solution**: The two-digit multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80 and 90. Of these, 20, 30, 40, 60, 80 and 90 are divisible by 15 or 20 (or both). This leaves just 10, 50 and 70. The two-digit multiples of 15 are 15, 30, 45, 60, 75 and 90. Of these, 30, 60 and 90 are multiples of 10 or 20, leaving just 15, 45 and 75. Finally, every multiple of 20 is also a multiple of 10, so none of our numbers can be divisible by 20. So the final list is 10, 15, 45, 50, 70, 75 and the final answer is **6**.

What is the difference between 9.99 and 99.9?

**Solution**: 99.9-9.99=**89.91**.

What is the smallest integer greater than 1 that can be written as a product of three identical integers, and can also be written as a product of four identical integers?

**Solution**: The answer is 16x16x16=8x8x8x8=**4096**.

How many minutes are there in one tenth of one third of one week?

**Solution**: There are 7x24x60 minutes in a week, so in one tenth of one third of this, there are 7x24x2=**336 **minutes.

On a farm, the ratio of cows to pigs is 4:5 and the ratio of cows to sheep is 3:4. There are no other animals. What fraction of all the animals are cows?

**Solution**: The ratio of cows to pigs is 12:15, and the ratio of cows to sheep is 12:16. So the ratio of cows to pigs to sheep is 12:15:16, and the fraction of animals that are cows is **12/43**

What is the mean of 5, half of 5, and half of half of 5?

**Solution**: Half of 5 is 2.5, and half of half of 5 is 1.25. So the required mean is (5+2.5+1.25)/3=8.75/3=**35/12**.