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? ?
Hint: First find the common denominator, then flip the result to find the reciprocal
Solution: Add 1/2 and 1/3 to finding the common denominator, then flip the result to find the reciprocal (just like turning the fraction upside down): 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?
Hint: Think about the smallest number that both 14 and 15 can divide evenly into (the least common multiple). Then think about the smaller numbers that divide into it (the factors)?
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.
The letters A, A, N, N are written on four tiles (one letter per tile) and placed in a bag. They are then drawn out of the bag at random, one at a time, and placed in a line from left to right. What is the probability that the tiles spell ANNA?
Hint: Start by thinking about the probability that the first tile you draw is an A. Then think about the probability of drawing two N’s in a row and what then is probability that the last letter will be A?
Solution: The probability that the first letter is A is 1/2. With an A removed, the probability the next two letters are both N’s is 2/3 x 1/2 = 1/3. With ANN removed, the last letter is always an A. So the probability of spelling ANNA is 1/2 x 1/3 = 1/6
What is the highest common factor of 117, 171 and 711?
Hint: Start by looking for a simple pattern, like the sum of the digits. Can each be divided by that number?
Solution: 117, 171 and 711 all have a digit sum of 9, so they are all multiples of 9. Dividing each by 9 leaves 13, 19 and 79, all of which are prime so have no common factors (other than 1). So the highest common factor is 9.
Insert two pairs of brackets into the expression 1-2-3-4 to make the result as large as possible.?
Hint: Think about how brackets change the order and remember that doing the maths inside the brackets first might change your answer.
Solution: The largest result is 6, so the best way to add brackets is: 1−((2−3)−4)
Divide 111,111,111 by 1,001,001 without a calculator.
Hint: Notice the patterns in the numbers
Solution: Here’s a cool trick! 1,001,001 can actually fit into 111,111,111 exactly 111 times. So: 111,111,111=1,001,001×111
To divide 111,111,111 by 1,001,001, you just reverse the multiplication: 111,111,111÷1,001,001=111. So, when you divide 111,111,111 by 1,001,001, the answer is 111!
How many square factors does 300 have?
Hint: Start by finding the factors of 300 (the numbers we can multiple together to get 300), then find the square numbers from these
Solution: The square factors of 300 are 1, 4, 25 and 100, so there are four factors
How many square numbers less than 1000 are also cube numbers?
Hint: A square number is when you multiply a number by itself. A cube number is when you multiply a number by itself twice. For example, 2×2×2=82 . Some numbers can be both a square and a cube. These numbers are special and are called sixth powers. This is because to be both, a number has to be like multiplying the same number six times. For example: 2×2×2×2×2×2=64, so 64 is both a square and a cube.
Solution: There are only three such numbers less than 1000: 1, 64 and 729
Two fifths of a number is 5. Find two thirds of the number.
Solution: One fifth of the number is 2.5, so the number is 5×2.5=12.5. Finally, two thirds of 12.5 is 2×12.5/3=25/3 or 8 1/3
A two-pence piece is glued to a table, with Heads facing up. Another two-pence piece is placed next to it, so that the two coins are touching. Then the second coin rolls around the first (fixed) coin without slipping, until it returns to its starting position. Through what angle does the second coin turn?
Solution: The second coin makes two complete revolutions (try it for yourself!), so it turns through 2×360=720 degrees.
Amber thinks of a number. She doubles it, then subtracts 7, then halves the result, ending up with -2. What number did she start with?
Solution: Amber starts with 1.5. She doubles it to get 3, subtracts 7 to get -4, then halves this to end up with -2. The answer is therefore 1.5
Samuel thinks of a number. He cubes it, then adds 30, then multiplies the result by -1, ending up with the number he first thought of. What is this number?
Solution: Samuel starts with -3. He cubes it to get -27, adds 30 to get 3, then multiplies this by -1 to get back to -3
The mean (average) height of a class of 9 pupils is 150cm. A 10th pupil then joins the class, increasing the average to 151.5cm. How tall is the 10th pupil?
Solution: The sum of the heights of the first 9 pupils is 150×9=1350cm, and the sum of the heights of all 10 pupils is 1515cm. So the 10th pupil is 1515-1350=165cm
There is a magic plant called the illusium. When an illusium is uprooted, 10 illusium plants grow in its place. In a large field there is a single illusium. No more illusiums will be planted in the field. Is it possible that, eventually, the field will contain exactly 999 illusiums?
Solution: It is not possible – every time an illusium is uprooted, the total number if illusiums in the field increases by exactly 9. So the number of illusiums is always 1 more than a multiple of 9; but 999 is itself a multiple of 9.
A palindrome is a number that reads the same forwards and backwards, like 77 and 30903. What is the largest three-digit palindrome divisible by 6?
Solution: A number is divisible by 6 if it is even and its digits sum to a multiple of 3. The largest three-digit palindrome with this property is 888
In how many ways can five people sit in a line on a bench, if two of the people, Alice and Bob, must sit together?
Solution: For the moment, treat Alice and Bob as a single two-headed person. Then there are 4 ‘people’ to arrange in a line, which can be done in 4x3x2x1=24 ways. For each of these 24 arrangements, there are two ways to arrange Alice and Bob – Alice can go to the right or to the left of Bob. So the total number of arrangements of all five people is 24×2=48
Divide 111,111,111 by 1,001,001 without a calculator.
Solution: We have 111,111,111=100,100,100+10,010,010+1,001,001=1,001,001(100+10+1)=1,001,001×111. So the answer is 111
Insert two pairs of brackets into the expression 1-2-3-4 to make the result as large as possible.
Solution: There are five ways to bracket 1-2-3-4. The way that produces the largest result is 1-((2-3)-4)=6
The letters A, A, N, N are written on four tiles (one letter per tile), and placed in a bag. They are then drawn out of the bag at random, one at a time, and placed in a line from left to right. What is the probability that the tiles spell ANNA?
Solution: The probability that the first letter is A is 1/2. With an A removed, the probability the next two letters are both N’s is 2/3 x 1/2 = 1/3. With ANN removed, the last letter is always an A. So the probability of spelling ANNA is 1/2 x 1/3 = 1/6
What is the highest common factor of 117, 171 and 711?
Solution: 117, 171 and 711 all have a digit sum of 9, so they are all multiples of 9. Dividing each by 9 leaves 13, 19 and 79, all of which are prime so have no common factors (other than 1). So the highest common factor is 9.
