time yourself

Maths A (word problems)Tue 15 September1 / 4

Multiples and factors, in words

You already have the tools. This week you only use them

Name Date

Last week, in one line

Lowest common multiple — when will they line up again.
Greatest common divisor — what is the largest equal share.

In a word problem, decide which one is being asked before you calculate.

Now you

1Bus A leaves the station every 9 minutes, bus B every 12. They have just left together. After how many minutes do they leave together again?

2In that time, how many buses does A send? And B?

Once you have both numbers, look at them carefully.

3A sheet of paper 24 cm by 36 cm is cut into identical squares with nothing left over. What is the largest possible side, and how many squares are there?

time yourself

Maths A (word problems)Tue 15 September2 / 4

Multiples and factors, in words

Name Date

4List every number from 1 to 100 that is a multiple of both 8 and 12.

542 pencils and 28 notebooks are shared equally with nothing left over. What is the greatest number of people, and what does each get?

A little further

6Find the lowest common multiple of 4, 6 and 9.

Three numbers work the same way as two.

7Two whole numbers have greatest common divisor 6 and lowest common multiple 72. Find them. There is more than one answer.

The product of two numbers equals GCD x LCM.

time yourself

Maths A (word problems)Tue 15 September3 / 4

Multiples and factors, in words

Name Date

8Check that 'product = GCD x LCM' holds, using 8 and 12.

9Find the greatest common divisor of 84 and 126.

With larger numbers, splitting into prime factors is quicker.

Cutting it up, or tiling it out

10At a station a train leaves every 8 minutes and a bus every 20. They left together at 7 am. Counting 7 am itself, how many times do they leave together before noon?

time yourself

Maths A (word problems)Tue 15 September4 / 4

Multiples and factors, in words

Name Date

11Rectangles 18 cm by 30 cm are laid out, all the same way round and with no gaps, to make a square. What is the side of the smallest such square, and how many rectangles does it take?

12Question 3 cut a large sheet into squares; question 11 tiled small sheets into a square. One of them needs the greatest common divisor and the other the lowest common multiple. Which is which, and why?

The important question today. Cutting up, against laying out.

Extra · you may skip this

Three gears (12, 18 and 24 teeth) mesh together. After how many teeth are all three back where they started?

Costume 1 · musicWed 16 September1 / 4

Chords and string length

This week's common denominators, turned into sound

Name Date

Think about this

Stop a guitar string exactly halfway and you hear the same note an octave higher, because the length became ¹⁄₂.

So what happens if you make the length ²⁄₃ instead?

1Write down what you think.

Stop the string at a different point and the note changes. The notes that sound good together are the ones whose lengths make simple fractions. Pythagoras found this 2,500 years ago, and this week's common denominators fall straight out of it.

Costume 1 · musicWed 16 September2 / 4

Chords and string length

Name Date

Now you — the string is 60 cm, and that note is C

2How long must the string be to sound C one octave higher?

3How long for G? (the length is ²⁄₃)

4How long for F? (the length is ³⁄₄)

5Which string is shorter, G or F? Compare using a common denominator.

²⁄₃ and ³⁄₄ cannot be compared as they stand.

Costume 1 · musicWed 16 September3 / 4

Chords and string length

Name Date

6A shorter string gives a higher note. Which is higher, G or F?

7On a 90 cm string, what length gives G?

8Explain in words why ²⁄₃ and ³⁄₄ cannot be compared as they are.

Put them in order

9E has length ⁴⁄₅. On the 60 cm string, how long is that?

Costume 1 · musicWed 16 September4 / 4

Chords and string length

Name Date

10Put ²⁄₃ (G), ³⁄₄ (F) and ⁴⁄₅ (E) in order, longest first.

Put them over 60 and the answer is the lengths in cm.

11Are the three you just ordered going from high notes to low, or low to high?

12Length ¹⁄₂ was C an octave up and ²⁄₃ was G. What note is length ¹⁄₃ ?

¹⁄₃ is ²⁄₃ of ¹⁄₂.

Extra · you may skip this

A 40 cm string sounded G. Halve it to 20 cm — what note do you get?

time yourself

Maths B (word problems)Fri 18 September1 / 4

Common denominators, in words

When you have an answer, always check whether it cancels

Name Date

The order

① find the LCM of the denominators　② multiply top and bottom to match　③ add or subtract the numerators only　④ check whether it cancels

Calculate

1²⁄₃ ＋ ¹⁄₄

2⁵⁄₈ − ¹⁄₆

3³⁄₁₀ ＋ ¹⁄₁₅

4⁵⁄₁₂ ＋ ¹⁄₄

This one cancels.

time yourself

Maths B (word problems)Fri 18 September2 / 4

Common denominators, in words

Name Date

Word problems — draw it before writing the calculation

5A ribbon is ⁷⁄₈ m long. ¹⁄₃ m is used. How much is left?

6There is ²⁄₅ L of juice and ³⁄₁₀ L more. How much altogether?

7Work out ¹⁄₂ ＋ ¹⁄₃ ＋ ¹⁄₄.

Three fractions still need only one common denominator.

time yourself

Maths B (word problems)Fri 18 September3 / 4

Common denominators, in words

Name Date

8Which is larger, ⁵⁄₇ or ⁷⁄₁₀ ? Use a common denominator.

9Someone wrote the answer to question 4 as ⁸⁄₁₂. It is not wrong, but it is not how we write it. Why?

A little further

10A tank holds ³⁄₄ L. ¹⁄₆ L is spilt, then ¹⁄₃ L is added. How much is in it now?

All three may go over one denominator at once.

time yourself

Maths B (word problems)Fri 18 September4 / 4

Common denominators, in words

Name Date

11Put ⁴⁄₉, ¹⁄₂ and ⁵⁄₁₂ in order, smallest first.

12Write every fraction with denominator 12 that is bigger than ¹⁄₃ and smaller than ¹⁄₂.

"Every" means how many there are is part of the answer.

Extra · you may skip this

Work out ¹⁄₂ ＋ ¹⁄₄ ＋ ¹⁄₈ ＋ ¹⁄₁₆. Look at the answer and write what you notice.

Costume 2 · chemistry + correctionsSun 20 September1 / 4

Balancing equations, and this week's corrections

A puzzle about making counts match. You do not need to know any chemistry

Name Date

Think about this

Put numbers in the boxes so that each kind of ball appears the same number of times on the left and on the right.

□H₂ ＋ O₂ → □H₂O

H₂ is two H balls, O₂ is two O balls, H₂O is two H and one O.

Start by counting the O balls.

1Fill in the boxes above.

The left has 2 O. Each H₂O on the right has 1.

It looks like chemistry, but all you are doing is making two counts match — the same shape as this week's common denominators.

Now you

2Fill in　CH₄ ＋ □O₂ → CO₂ ＋ □H₂O

Count C, then H, then O, and you will not get lost.

Costume 2 · chemistry + correctionsSun 20 September2 / 4

Balancing equations, and this week's corrections

Name Date

3Fill in　□N₂ ＋ □H₂ → □NH₃

4Fill in　□Fe ＋ □O₂ → □Fe₂O₃

O comes in 2s on the left and 3s on the right. Think about the lowest common multiple of 2 and 3.

5Explain in words why a lowest common multiple turns up in question 4.

This week's corrections

6Pick two questions you got wrong this week and do them again.

Solve them again — do not copy the answer across.

Costume 2 · chemistry + correctionsSun 20 September3 / 4

Balancing equations, and this week's corrections

Name Date

Classic problem — planting trees

7Trees are planted along a straight road every 4 m, from one end to the other. The road is 60 m long. How many trees are needed?

60 ÷ 4 = 15, but 15 is not the answer. Draw it.

More of it — a straight road, and a closed loop

8Along the same 60 m road, trees every 5 m. How many trees?

9Round a circular pond, trees every 4 m. If the perimeter is 60 m, how many trees?

Costume 2 · chemistry + correctionsSun 20 September4 / 4

Balancing equations, and this week's corrections

Name Date

10Question 7 and question 9 are both 60 m at 4 m intervals, and the answers differ by one. Why? Explain in words.

The important question today.

11Along both sides of a 48 m road, trees every 6 m. How many altogether?

12Of the whole numbers 1 to 100, how many are multiples of neither 3 nor 4?

Counting multiples of 3 and of 4 counts the multiples of 12 twice.

Extra · you may skip this

To plant 21 trees along that same 60 m road, how far apart should they be?