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Maths

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Maths Wizard

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Hello young Maths Wizards,

Did you enjoy solving my mathematical conundrums last week? I have 4 more for you to try!

The Maths Wizard will be going on his summer holiday now, and will be back with new maths problems on 12 August. We hope you enjoy these mathematical challenges and have a relaxing summer break!

Remember to check out my Previous Questions and the Further Resources tabs if you’d like more mathematical challenges!

Don’t forget you can check your answers on the Maths Wizard Answer tab, but no peeking until you’ve done your best!

So if you’re ready, click on my wand for this week’s challenge!

Good Luck!

1.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?

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

3. On 27/09/21 the sum of the digits of the day (2+7) was equal to the number of the month (9). How many more times will this happen in 2021?

4. Does there exist a number that is larger than its cube?

22 July 2022

Hello young Maths Wizards,

Did you enjoy solving my mathematical conundrums last week? I have 4 more for you to try!

The Maths Wizard will be going on his summer holiday now, and will be back with new maths problems on 12 August. We hope you enjoy these mathematical challenges and have a relaxing summer break!

Remember to check out my Previous Questions and the Further Resources tabs if you’d like more mathematical challenges!

Don’t forget you can check your answers on the Maths Wizard Answer tab, but no peeking until you’ve done your best!

So if you’re ready, click on my wand for this week’s challenge!

Good Luck!

Here are the answers to this week’s tricky mathematical conundrums:

1.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?

Answer: 6/5

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

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

Answer: 14

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.

3. On 27/09/21 the sum of the digits of the day (2+7) was equal to the number of the month (9). How many more times will this happen in 2021?

Answer: three more times

Solution: This will happen again on 19/10, 28/10 and 29/11; that is three more times.

4. Does there exist a number that is larger than its cube?

Answer: yes

Solution: There exist lots of numbers larger than their cube: any number between 0 and 1, and any number less than -1; so the answer is yes.

Hello young Maths Wizards,

Did you enjoy solving my mathematical conundrums last week? I have 4 more for you to try!

The Maths Wizard will be going on his summer holiday now, and will be back with new maths problems on 12 August. We hope you enjoy these mathematical challenges and have a relaxing summer break!

Remember to check out my Previous Questions and the Further Resources tabs if you’d like more mathematical challenges!

Don’t forget you can check your answers on the Maths Wizard Answer tab, but no peeking until you’ve done your best!

So if you’re ready, click on my wand for this week’s challenge!

Good Luck!

In a non-leap year, what is the date of the middle day of the year?

Answer: 2nd July

Solution: There are 365 days in a non-leap year, so the middle day is day 183. There are 31+28+31+30+31+30=181 days in the first six months; so the middle day is 2nd July.

 

Convert 10,000 minutes into days, hours and minutes.

Answer: 6 days, 22 hours and 40 minutes

Solution: There are 60 minutes in 1 hour, and 24×60=1440 minutes in a day. So there are 6×1440=8640 minutes in 6 days, leaving 10,000-8640=1360 minutes. This is 80 minutes less than 1 day. So 10,000 minutes is equivalent to 6 days, 22 hours and 40 minutes.

 

How many factors of 333 contain at least one digit 3?

Answer: 3

Solution: The factors of 333 containing a digit 3 are 3, 37 and 333 itself; that is three factors.

 

Which positive integer(s) below 100 has/have the most factors containing at least one digit 3?

Answer: 39, 78 & 93

Solution: There are three positive integers below 100 that have the most factors containing a digit 3: 39, 78 and 93 each have exactly three factors containing a digit 3.

 

In how many ways is it possible to pick three whole number angles to make an isosceles triangle?

Answer: 89

Solution: The base angles are equal, and can take any whole number value from 1 degree to 89 degrees, so this can be done in 89

 

I have a circle made of paper. What is the smallest number of folds required to turn the circle into a polygon (a shape with straight edges)?

Answer: 3

Solutions: Every edge requires a different fold, and every polygon has at least three sides, so the best we can hope for is three. And it is in fact possible to create a triangle with three folds; so the answer is three.

 

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?

Answer: 1/6

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?

 Answer: 9

Solutions: 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.

 

How many positive numbers are factors of both 45 and 60?

Answer: 4

Solution: The common factors of 45 and 60 are precisely the factors of 15, namely 1, 3, 5 and 15. So the answer is 4.

 

How many positive numbers are factors of 100 but not 50?

 Answer: 3

Solutions: We require those factors of 100 that are divisible by 4, namely 4, 20 and 100, So the answer is 3

 

Divide 111,111,111 by 1,001,001 without a calculator

Answer: 111

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

Answer: 1-((2-3)-4)=6

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 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?

Answer: 165cm

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 tall

 

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?

Answer: It is not possible

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.

How many square factors does 300 have?

Answer: 4

Solution: The square factors of 300 are 1, 4, 25 and 100 – that is four

 

How many square numbers less than 1000 are also cube numbers?

Answer: 3

Solutions: Squares that are also cubes must be sixth powers (some integer raised to the power of six). 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.

Answer: 25/3 or 8 1/3

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?

Answer: 720 degrees

Solution: The second coin makes two complete revolutions (try it for yourself!), so it turns through 2×360 = 720 degrees

 

Daria thinks of a number. She doubles it, then subtracts 7, then halves the result, ending up with -2. What number did she start with?

Answer: 1.5

Solution: Daria starts with 1.5. She doubles it to get 3, subtracts 7 to get -4, then halves this to end up with -2.

 

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?

Answer: -3

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

 

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?

Answer: 888

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?

Answer: 48

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

A fish weighs 3kg plus a third of its weight. How much does the fish weigh?

Answer: 4.5kg

Solution: One third plus two thirds makes a whole, so 3kg is two thirds of the weight. Hence 1.5kg is one third of the weight, and the total weight of the fish is 1.5×3=4.5kg

 

In an isosceles triangle, one of the angles is 40 degrees, and one of the other angles is x degrees. What are the possible values of x?

Answer: 40, 70 and 100

Solution: If 40 degrees is one of the two base angles, then the other two angles are 40 degrees and 100 degrees. If, however, 40 degrees is the angle at the apex, then the two base angles are equal to 70 degrees. So the possible values of x are 40, 70 and 100

 

In how many ways can 60 be written as a sum of consecutive positive odd numbers?

Answer: Two

Solution: Two (5+7+9+11+13+15 and 29+31).

 

Suppose all the arrangements of the letters in the word FACTOR are listed in dictionary order (so the list begins 1 ACFORT, 2 ACFOTR, 3 ACFROT, …). In what position does the word FACTOR appear?

Answer: the 245thposition

Solution: There are 6x5x4x3x2x1=720 arrangements of the letters in the word FACTOR. The first 120 of them begin with an A; the next 120 begin with a C. Then we have 241 FACORT, 242 FACOTR, 243 FACROT, 244 FACRTO, and 245 FACTOR. So FACTOR appears in the 245th

 

Monica cuts a cake into 10 large pieces. She then takes half of these and cuts them into 3 medium pieces each. Finally, she takes a third of the medium pieces and cuts each of them into 2 small pieces.

(i) How many pieces of cake are there at the end?

Answer: 25

Solution: Of the 10 large pieces, 5 are left alone. The other 5 are cut into 15 medium pieces. Of these 15 pieces, 10 are left alone. The other 5 are cut into 10 small pieces. So at the end, there are 5 large, 10 medium and 10 small slices; that is 25 pieces in total.

(ii) Josh takes one piece of each size. What fraction of the original cake does this represent?

Answer: 1/6

Solution: Each large piece represents 1/10 of the original cake. Each medium piece represents 1/30 of the original cake. Each small piece represents 1/60 of the original cake. So the fraction represented by Josh’s portion is (1/10)+(1/30)+(1/60)=(6/60)+(2/60)+(1/60)=(10/60)=1/6 (One sixth).

 

Amelia is three times as old as Jay. In four years, Amelia will be twice as old as Jay. How many years older than Jay is Amelia?

Answer: 8

Solution: 8 years – Amelia is currently 12, Jay is currently 4

 

There is a circular table with six empty seats around it. Charlie and Dilshan pick different seats at random. What is the probability they sit opposite each other?

Answer: 1/5

Solution: Wherever Charlie sits, there will be five seats remaining for Dilshan, only one of which is opposite Charlie. So the answer is 1/5

 

How many primes less than 200 contain a digit 0?

Answer: 4

Solution: 4 (101, 103, 107, 109)

 

A rectangular grid of squares is coloured blue. A border, one square wide, is constructed around the rectangular grid, and painted yellow. The number of blue squares is equal to the number of yellow squares. Another rectangular grid of squares with different dimensions is also coloured blue, and a similar yellow border is constructed. The number of blue squares in this grid is equal to the number of yellow squares comprising its border. How many squares are used in total in both grids combined?

Answer: 108

Solution: 108 (One grid has dimensions of 3×10, the other 4×6)

 

Annie thinks of a number. She triples it, then subtracts 10, ending up with -25. What was her original number?

Answer: -5

Solution:  Reverse the operations Annie carried out: add 10 to -25 to obtain -15, then divided -15 by 3 to obtain -5

 

What is the smallest number divisible by exactly one prime number and exactly six other positive integers?

Answer: 64

Solution:  Any such number will be of the form p x p x p x p x p x p for some prime number p. The smallest number of this type is 2x2x2x2x2x2=64 (Its factors are 1, 2, 4, 8, 16, 32 and 64)

 

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

Answer: 894

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.

 

Hello young Maths Wizards,

Did you enjoy solving my mathematical conundrums last week? I have 4 more for you to try!

The Maths Wizard will be going on his summer holiday now, and will be back with new maths problems on 12 August. We hope you enjoy these mathematical challenges and have a relaxing summer break!

Remember to check out my Previous Questions and the Further Resources tabs if you’d like more mathematical challenges!

Don’t forget you can check your answers on the Maths Wizard Answer tab, but no peeking until you’ve done your best!

So if you’re ready, click on my wand for this week’s challenge!

Good Luck!

If you really enjoy solving mathematical problems or if you’d just like a bit more practice, here are some web sites you can look at. I’m certain you will find plenty to challenge you!

Click on any of the links to take you straight there.

Junior Maths Challenge

Here you can try online challenges from previous Junior Maths Challenges, as well as download past papers and solutions.

UKMT Junior Maths Challenge

 

Nrich Website / Twitter

The NRICH Project aims to enrich the mathematical experiences of all learners. They have lots of activities, questions and games to develop your mathematical skills, whatever level you are. They definitely make maths fun!

NRICH Primary Maths

They also post questions on twitter for all ages of students.

NRICH Twitter Account

 

BBC Bitesize

Lots of fun problem solving questions for KS2.

BBC Bitesize Maths

 

Times Tables Rockstars

Times Tables Rock Stars is a carefully sequenced programme of fun daily times tables practice, which concentrates on a different times table each week. It has a small subscription of £7.20 a year for a family.

Times Tables Rockstars

 

As well as these web sites, I can also recommend the brilliant maths book Elastic Numbers, written by one of Hampton’s own Maths Teachers, Mr Griller.

Elastic Numbers is full of fun and challenging mathematical treats for the serious problem solvers among you! It’s available from most book shops and also online.

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