Tag Archives: short cuts

Tips for the QTS numeracy test

“If I’d known I’d have to go back to school, I’d never have become a teacher!” Back-to-school-blackboard-chalk

The QTS numeracy and literacy tests are not very popular, but trainee teachers still have to pass them before they can start teaching in the state sector, so I thought I’d try and help out. There is always more than one way of doing a Maths question, but I hope I’ll demonstrate a few useful short cuts and describe when and how they should be used. The point of short cuts is that, even though you may have to do more sums, they’ll be easier sums that can be done faster and more accurately. The numeracy test consists of two sections – mental Maths and interpreting charts – and I’m going to focus on the first of these.

Fractions to percentages – type 1

There are a number of typical types of questions in the numeracy test, and a lot of them involve multiplication – so knowing your times tables is an absolute must! One of the most common kinds of question involves converting fractions to percentages. These are just two ways of showing the same thing, but to answer these questions you’ll need to try different approaches. First of all, have a look to see if the denominator (or the number on the bottom of the fraction) is a factor or a multiple of 100. If it is, you can simply multiply or divide the numerator (the number on the top) and the denominator by whatever it takes to leave 100 on the bottom. Any fraction over 100 is just a percentage in disguise, so you just need to put the percentage sign after the numerator, eg what is the percentage mark if:

  1. a pupil scores 7 out of a possible 20?
    Answer: 20 x 5 = 100, so 7 x 5 = 35%.
  2. a pupil scores 18 out of a possible 25?
  3. a pupil scores 7 out of a possible 10?
  4. a pupil scores 9 out of a possible 20?
  5. a pupil scores 130 out of a possible 200?

Fractions to percentages – type 2

If the denominator is not a factor of 100, check if it’s a multiple of 10. If it is, you can convert the fraction into tenths and then multiply the top and bottom by 10 to get a fraction over 100, which, again, is just a percentage in disguise, eg what is the percentage mark if:

  1. A pupil scores 24 marks out of a possible 40?
    Answer: 40 ÷ 4 = 10, so 24 ÷ 4 = 6 and 6 x 10 = 60%.
  2. A pupil scores 12 marks out of a possible 30?
  3. A pupil scores 32 marks out of a possible 80?
  4. A pupil scores 49 marks out of a possible 70?
  5. A pupil scores 24 marks out of a possible 60?

Fractions to percentages – type 3

If neither of the first two methods works, that means you have to simplify the fraction. Once you’ve done that, you should be able to convert any common fraction into a percentage in your head. The most commonly used fractions are halves, quarters, fifths and eighths, so it’s worth learning the decimal and percentage equivalents off-by-heart, ie

  • ½ = 0.5 = 50%
  • ¼ = 0.25 = 25%
  • ¾ = 0.75 = 75%
  • ⅕ = 0.2 = 20%
  • ⅖ = 0.4 = 40%
  • ⅗ = 0.6 = 60%
  • ⅘ = 0.8 = 80%
  • ⅛ = 0.125 = 12.5%
  • ⅜ = 0.375 = 37.5%
  • ⅝ = 0.625 = 62.5%
  • ⅞ = 0.875 = 87.5%

To simplify the fractions, check first to see if the numerator goes into the denominator. If it does, you can simply divide both numbers by the numerator to get what’s called a ‘unit fraction’, in other words a fraction with a one on top. By definition, a unit fraction can’t be simplified, so then you just have to convert it into a percentage. If the numerator doesn’t go exactly, try the first few prime numbers, ie 2, 3, 5, 7 and perhaps 11. Keep dividing both numbers in the fraction by the lowest possible prime number, and you’ll eventually show the fraction in its lowest terms. (If you happen to find a bigger number you can use, that’s great, as it means you won’t need to do as many sums.) When you’re left with one of the common fractions in the list above, you just have to convert it into the correct percentage, eg what is the percentage mark if:

  1. a pupil scores 7 out of a possible 28?
    Answer: 7 goes into 28 four times, so the fraction is 1/4, which is 25%.
  2. a pupil scores 27 out of a possible 36?
    Answer: 27 doesn’t go into 36, but 3 does, so 27/36 = 9/12, but 9 and 12 are also divisible by 3, so that makes 3/4, which is 75%.
  3. a pupil scores 24 out of a possible 48?
  4. a pupil scores 8 out of possible 32?
  5. a pupil scores 9 out of a possible 24?

Multiplying three numbers involving money

There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it, eg 1 x 2 x 3 is the same as 3 x 2 x 1. The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning and then multiply the remaining three numbers together, eg a number of pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get a certain number of correct spellings, and the average amount of sponsorship is shown for each. How many pounds would the class expect to raise for charity if the basic sum is:

  1. 20 x 30 x 5p?
    Answer: 2 x 3 x 5 = 6 x 5 = £30.
  2. 40 x 500 x 7p?
  3. 30 x 400 x 6p?
  4. 50 x 40 x 8p?
  5. 60 x 20 x 9p?

Division by single-digit numbers

This is what I call the ‘wedding planner problem’. There are three ways of doing this type of question:

  • Method A: Use the ‘bus stop’ method to divide the total number of guests by the number of seats per table – remembering to add one if there is a remainder.
  • Method B: Go straight to the end of your times tables by multiplying the number of seats by 12, then calculating the remainder and dividing by the number of seats per table, again remembering to add one if there is another remainder.
  • Method C: Use trial and error by estimating the number of tables needed using a nice, round number such as 5, 10 or 20 and working out the remainder as before.
  1. Dining tables seat 7 children. How many tables are needed to seat 100 children?
    Answer:
    Method A) 100 ÷ 7 = 14 r 2, so 14 + 1 = 15 tables are needed.

    Method B) 7 x 12 = 84, 100 – 84 = 16, 16 ÷ 7 = 2 remainder 2, 12 + 2 + 1 = 15 tables.
    Method C) 10 x 7 = 70, which is too small, 20 x 7 = 140, which is too big, 15 x 7 = 70 + 35 = 105, which is just right as there are only 5 seats to spare.
  2. Dining tables seat 6 children. How many tables are needed to seat 92 children?
  3. Dining tables seat 5 children. How many tables are need to seat 78 children?
  4. Dining tables seat 9 children. How many tables are needed to seat 120 children?
  5. Dining tables seat 6 children. How many tables are needed to seat 75 children?

Division by two-digit numbers

If the number of seats is outside your times tables, the best option is just to use trial and error, starting with 5, 10 or 20, eg

  1. It is possible to seat 40 people in a row across the hall. How many rows are needed to seat 432 people?
    Answer: 40 x 10 = 400, 432 – 400 = 32, so one more row is needed, making a total of 10 + 1 = 11 rows.
  2. It is possible to seat 32 people in a row across the hall. How many rows are needed to seat 340 people?
  3. It is possible to seat 64 people in a row across the hall. How many rows are needed to 663 people?
  4. It is possible to seat 28 people in a row across the hall. How many rows are needed to seat 438 people?
  5. It is possible to seat 42 people in a row across the hall. How many rows are needed to seat 379 people?

Percentages to fractions

This is a type of question that looks hard at first but becomes dead easy with the right short cut. All you need to do is to work out 10% first and then multiply by the number of tens in the percentage. Another way of saying that is just to knock one zero off each number and multiply them together, eg a test has a certain number of questions, each worth one mark. For the stated pass mark, how many questions had to be answered correctly to pass the test?

  1. ?/30 = 40%
    Answer: 3 x 4 = 12 questions (ie 10% of 30 is 3 questions, but we need 40%, which is 4 x 10%, so we need four lots of three, which is the same as 3 x 4).
  2. ?/40 = 70%
  3. ?/50 = 90%
  4. ?/80 = 70%
  5. ?/300 = 60%

Ratio – distance

There are two ways of converting between different units of distance from the metric and imperial systems:

  • Method A: Make the ratio into a fraction and multiply the distance you need to find out by that same fraction, ie multiply it by the numerator and divide it by the denominator. (Start with multiplication if doing the division first wouldn’t give you a whole number.)
  • Method B: Draw the numbers in a little 2 x 2 table, with the figures in the ratio in the top row and the distance you need to find out in the column with the appropriate units, then find out what you need to multiply by to get from the top row to the bottom row and multiply the distance you have to find out by that number to fill in the final box.
  1. 8km is about 5 miles. How many kilometres is 40 miles?
    Answer:
    Method A) 8:5 becomes 8/5, and 40 x 8/5 = 40 ÷ 5 x 8 = 8 x 8 = 64km.
    Method B)
    Miles                 km
    5                          8
    x 8
    40               8 x 8 = 64km
  2. 6km is about 4 miles. How many kilometres is 36 miles?
  3. 4km is about 3 miles. How many kilometres is 27 miles?
  4. 9km is about 7 miles. How many miles is 63 kilometres?
  5. 7km is about 4 miles. How many kilometres is 32 miles?

Ratio – money

You can use the same methods when converting money, except that the exchange rate is now a decimal rather than a fraction. Just remember that the pound is stronger than any other major currency, so there will always be fewer of them. It’s easy to get things the wrong way round, so it’s worth spending a couple of seconds checking, eg

  1. £1 = €1.70. How much is £100 in euros?
    Method A) 100 x 1.70 = €170.
    Method B)
    £                                      €
    1.00                              1.70
    x 100
    100                    1.70 x 100 = €170
  2. £1 = €1.60. How much is £200 in euros?
  3. £1 = €1.50. How much is €150 in pounds?
  4. £1 = €1.80. How much is €90 in pounds?
  5. £2 = €3.20. How much is £400 in euros?

Time – find the end time

The most useful trick to use here is rounding. If the length of a lesson is 45 minutes or more, then just round up to the full hour and take the extra minutes off at the end. This avoids having to add or subtract ‘through the hour’, which is more difficult. If the lessons are less than 45 minutes long, just work out the total number of minutes, then convert into hours and minutes and add to the start time, eg

  1. A class starts at 9:35. The class lasts 45 minutes. What time does the class finish?
    Answer: 9:35 + 1 hour – 15 minutes = 10:35 – 15 minutes = 10:20.
  2. A class starts at 11:45. There are three consecutive classes each lasting 25 minutes and then half an hour for lunch. What time does lunch finish?
    Answer: 11:45 + 3 x 25 + 30 = 11:45 + 75 + 30 = 11:45 + 1 hour and 15 minutes + 30 minutes = 13:30.
  3. Lessons start at 11:15. There are two classes each lasting 40 minutes and then lunch. What time does lunch start?
  4. Lessons start at 2:00 in the afternoon. There are four 50-minute classes with a 15-minute break in the middle. What time does the day finish?
  5. Lessons start at 9:40. There are two classes of 50 minutes each with a break of 15 minutes in between. What time do the classes finish?

Time – find the start time

It’s even more important to use rounding when working backwards from the end of an event, as subtraction is that bit more difficult, eg

  1. A school day finishes at 3:15. There are two classes of 50 minutes each after lunch with a break of 15 minutes in the middle. What time does lunch end?
    Answer: 3:15 – 2 hours + 2 x 10 minutes – 15 minutes = 1:15 + 20 minutes -15 minutes = 1:20.
  2. A school day finishes at 4:30. There are two classes of 40 minutes each after lunch. What time does lunch finish?
    Answer: 4:30 – 2 x 40 = 4:30 – 80 minutes = 4:30 – 1 hour and 20 minutes = 3:10.
  3. Lunch starts at 1:05. There are two classes before lunch of 55 minutes each. What time do the classes start?
  4. Lunch starts at 1:15. There are three classes before lunch of 45 minutes each. What time do the classes start?
  5. A school bus arrives at school at 8:45. It picks up 20 children, and it takes an average of four minutes to pick up each child. What time is the first child picked up?

Percentage to decimal

A decimal is a fraction of one unit, but a percentage is a fraction of 100 units, so, to convert from a percentage to a decimal, you just need to divide by 100, eg

  1. What is 20% as a decimal?
    Answer: 20 ÷ 100 = 0.2.
  2. What is 30% as a decimal?
  3. What is 17% as a decimal?
  4. What is 6% as a decimal?
  5. What is 48% as a decimal?

Multiplying decimals

Decimal points can be confusing, so the best way to do these sums is to take out the decimal point and put it back at the end. You just need to remember to make sure there are the same number of decimal places in the answer as in both numbers in the question, eg

  1. 1.5 x 1.5
    Answer: 15 x 15 = 10 x 15 + 5 x 15 = 150 + 75 = 225, but there are two decimal places in the numbers you’re multiplying together, so the answer must be 2.25.
  2. 3 x 4.5
  3. 4.7 x 8
  4. 7.5 x 7.5
  5. 2.5 x 6.5

Multiplying decimals by a power of 10

Because we have 10 fingers, we’ve ended up with a ‘decimal’ number system based on the number 10. That makes it really easy to multiply by powers of 10, because all you have to do is to move the decimal point to the right by a suitable number of places, eg one place when multiplying by 10, two when multiplying by 100 etc. (You can also think of it as moving the digits in the opposite direction.) This type of question is therefore one of the easiest, eg

  1. 4.5 x 10
    Answer: 45.
  2. 3.8 x 100
  3. 7.6 x 1000
  4. 4.6 x 100
  5. 3.5 x 10

Percentage of quantity

Finding a percentage is easy if it ends with a zero, as you can start by finding 10% (Method A). If you happen to know what the fraction is, you can also divide by the numerator of that fraction (Method B), so 20% is 1/5, so you just need to divide by five, eg

  1. Find 20% of 360
    Answer:
    Method A) 360/10 x 2 = 36 x 2 = 72.
    Method B) 360 ÷ 5 = 72 (or 360 x 2 ÷ 10 = 720 ÷ 10 = 72).
  2. Find 20% of 45
  3. Find 30% of 320
  4. Find 60% of 60
  5. Find 80% of 120

Multiplication

Just because this is the ‘mental Maths’ section of the test doesn’t mean that you can’t work things out on paper, and these simple multiplication sums can be done like that. Alternatively, you can use ‘chunking’, which means multiplying the tens and units separately and adding the results together, and the short cut for multiplying by five is to multiply by 10 and then divide by two, eg

  1. 23 x 7
    Answer: 20 x 7 + 3 x 7 = 140 + 21 = 161.
  2. 42 x 5
    Answer: 42 x 10 ÷ 2 = 420 ÷ 2 = 210
  3. 34 x 6
  4. 56 x 8
  5. 34 x 8

Short division

Again, working these sums out on paper is probably quicker (and more reliable), although the easiest way to divide by four is probably to halve the number twice, and the short cut for dividing by five is to multiply by two and then divide by 10.

  1. 292 ÷ 4
    Answer: 292 ÷ 2 ÷ 2 = 146 ÷ 2 = 73.
  2. 345 ÷ 5
    Answer: 345 x 2 ÷ 10 = 690 ÷ 10 = 69.
  3. 282 ÷ 3
  4. 565 ÷ 5
  5. 432 ÷ 4
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Short cuts

Short cuts

It’s just through here…

There is always more than one way of solving a Maths problem. That can be confusing, but it can also be an opportunity – if only you can find the right trade-off between speed and accuracy. I’ve taught a lot of QTS numeracy candidates recently, and the Maths itself isn’t particularly difficult, particularly in the mental arithmetic section. The trick is to be familiar with all the possible short cuts and capable of using the right one at the right time. It may mean having to do more sums, but it will be much simpler and quicker in the long run. You don’t have to use all of these all the time, but it is useful to know what they are just in case you need them.

  • Multiplying and dividing by 5
    The most useful short cut I’ve come across is very simple. To multiply by 5, try multiplying by 10 and then dividing by 2 (or vice versa), eg
    13 x 5
    = 13 x 10 ÷ 2
    = 130 ÷ 2
    = 65
    You have to do two sums rather than one, but the point is that you should be able to save time and improve the chances of getting the answer right by doing both in your head rather than having to work out a more difficult sum on paper.
    You can do divide by 5 in a similar way by multiplying by 2 and dividing by 10 (or vice versa), eg
    65 ÷ 5
    = 65 x 2 ÷ 10
    = 130 ÷ 10
    = 13
    You can do a similar trick with 50, 500 etc simply by multiplying or dividing by a higher power of ten.
  • Chunking
    If you have to multiply by a two-digit number outside your times tables, chunking is an easy way to do the sum in your head. Instead of writing it down on paper and using long multiplication (which would take a long time and is easy to get wrong!), try multiplying by the tens and the units separately and adding up the results, eg 16 x 15 = 10 x 15 + 6 x 15 = 150 + 90 = 240. The numbers might still be too tricky to do it comfortably, but it’s often worth a try.
  • Rounding
    To avoid sums with ‘tricky’ numbers, try rounding them up to the nearest ‘easy’ figure and adjusting at the end. This is particularly useful when working out start and end times, eg
    ‘The morning session in a school began at 09:25. There were three lessons of 50
    minutes each and one break of 20 minutes. At what time did the morning session end? Give your answer using the 24-hour clock.’
    If you assume the lessons last an hour, you can add three hours to 09:25 to get 12:25. You would normally then knock off 3 x 10 = 30 minutes, but the 20-minute break means you only need to subtract 10 minutes, which means the session ended at 12:15.
  • Money problems
    There is often a ‘real world’ money problem in the QTS numeracy test. That usually means multiplying three numbers together. The first thing to say is that it doesn’t matter in which order you do it – 1 x 2 x 3 is just the same as 3 x 2 x 1. The next thing to bear in mind is that you will usually have to convert from pence to pounds. You could do this at the end by simply dividing the answer by 100, but a better way is to divide one of the numbers by 100 (or two of the numbers by 10) at the beginning or turn multiplication by a fraction of a pound into a division sum, eg
    ‘All 30 pupils in a class took part in a sponsored spell to raise money for charity. The pupils were expected to get an average of 18 spellings correct each. The average amount of sponsorship was 20 pence for each correct spelling. How many pounds would the class expect to raise for charity?’
    The basic sum is 30 x 18 x 20p, and there are a couple of ways you could do this:
    1) Knock off the zeroes in two of the numbers, change the order of the numbers to make it easier and double and halve the last pair to give yourself a sum in your times tables, ie
    30 x 18 x 20p
    = 3 x 18 x 2
    = 3 x 2 x 18
    = 6 x 18
    = 12 x 9
    = £108
    2) Convert pence into pounds, turn it into a fraction, change the order of the numbers, divide by the denominator and, again, double and halve the last pair to give yourself a sum in your times tables, ie
    30 x 18 x 20p
    = 30 x 18 x £0.20
    = 30 x 18 x ⅕
    = 30 x 18 ÷ 5
    = 30 ÷ 5 x 18
    = 6 x 18
    = 12 x 9
    = £108
  • Percentages
    Many students get intimidated by percentages, fractions and decimals, but they are all just different ways of showing what share you have of something. You will often by asked to add or subtract a certain percentage. The percentage will usually end in zero (eg 20%, 30% or 40%), so the easiest way is probably to find 10% first. That just means dividing by 10, which means moving the decimal point one place to the left or, if you can, knocking off a zero. Once you know what 10% is, you can simply multiply by 2, 3 or 4 etc and add or subtract that number to find the answer, eg
    ‘As part of the numeracy work in a lesson, pupils were asked to stretch a spring to extend its length by 40 per cent. The original length of the spring was 45 centimetres. What should be the length of the extended spring? Give your answer in centimetres.’
    You need to find 40% of 45cm, so you can start by finding 10%, which is 45 ÷ 10 or 4.5cm. You can then multiply it by 4 to find 40%, which is best done by doubling twice, ie 4.5 x 2 x 2 = 9 x 2 = 18. Finally, you just add 18cm to the original length of the spring to find the answer, which is 45 + 18 = 63cm.
  • Common fractions
    An awful lot of questions involve converting between fractions, percentages and decimals. There is a proper technique for doing any of those, but it’s very useful if you learn the most common fractions and their decimal and percentage equivalents by heart, eg
    ½ = 0.5 = 50%
    ¼ = 0.25 = 25%
    ¾ = 0.75 = 75%
    ⅕ = 0.2 = 20%
    ⅖ = 0.4 = 40%
    ⅗ = 0.6 = 60%
    ⅘ = 0.8 = 80%
    ⅛ = 0.125 = 12.5%
    ⅜ = 0.375 = 37.5%
    ⅝ = 0.625 = 62.5%
    ⅞ = 0.875 = 87.5%
  • Times tables
    There are far more multiplication questions in the QTS numeracy test than any other kind, so it’s very important to know your times tables inside out. Some pupils are taught to memorise only the results, eg 4, 8, 12… etc. This is catastrophic! If you have to go through the whole table to find the answer, counting off the number of fours on your fingers, you can’t save yourself any time at all. The proper way is to learn the whole sum, eg 1 x 4 is 4, 2 x 4 is 8, etc (or 1 4 is 4, 2 4s are 8, etc). That way, the answer to any question in your times tables will pop into your head as soon as you’ve heard it. One good way of learning your tables is to time yourself using the stopwatch function on your iPhone. If you press ‘Lap’ after you’ve recited each table, you can write down your times and work out which tables you need to practise. Once you’re confident, you can make certain sums fit into your times tables by doubling one number and halving the other, eg
    3 x 24
    = (3 x 2) x (24 ÷ 2)
    = 6 x 12
    = 72
    Alternatively, you can halve just one of the numbers and double the result, eg
    24 x 9
    = 12 x 9 x 2
    = 108 x 2
    = 216
  • Multiplying by 4
    If you have to multiply by 4 and the number is not in your times tables, a simple way to do it is to double it twice, eg
    26 x 4
    = 26 x 2 x 2
    = 52 x 2
    = 104
  • Multiplying by a multiple of 10
    If you have to multiply by a multiple of 10 such as 20 or 30, try knocking the zero off and adding it in again afterwards. That way, you don’t have to do any long multiplication and, with any luck, the sum will be in your times tables, eg
    12 x 30
    = 12 x 3 x 10
    = 36 x 10
    = 360
  • Multiplying decimals
    This can be a bit confusing, so the best way of doing it is probably to ignore any decimal points, multiply the numbers together and then add back the decimal point to the answer so that you end up with the same number of decimal places as you had in the beginning, eg
    0.5 x 0.5
    = 5 x 5 ÷ 100
    = 25 ÷ 100
    = 0.25
  • Using the online calculator
    The second section of the QTS numeracy test consists of on-screen questions that can be answered using an online calculator. This obviously makes working out the answer a lot easier, and short cuts are therefore less useful. However, just because the calculator’s there doesn’t mean you have to use it, particularly for multiple-choice questions. If you have to add up a column of cash values, for example, and compare it with a number of options, you could simply tot up the number of pence and pick the option with the right amount. Alternatively, the level of accuracy needed in the answer may give you a helping hand if it rules out all but one of the possible answers, eg 6 ÷ 21 to one decimal place is always going to be 0.3. Why? Well, it’s a bit less than 7 ÷ 21, which would be a third or 0.3 recurring. An answer of 0.4 would be more than that, and 0.2 would be a fifth, which is far too small, so it must be 0.3.
  • Don’t do more than you have to!
    There are several types of question that could tempt you into doing more work than you need to do. If you’re trying to work out how many tables you need at a wedding reception for a given number of guests, the answer is always going to need rounding up to the next whole number, so you don’t need to spend any time working out the exact answer to one or two decimal places. Equally, some numbers are so close to being an ‘easy’ number that you don’t need to add or subtract anything after rounding up or down to make the basic sum easier, eg
    ‘For a science experiment a teacher needed 95 cubic centimetres of vinegar for each pupil. There were 20 pupils in the class. Vinegar comes in 1000 cubic centimetre
    bottles. How many bottles of vinegar were needed?
    If you round 95cc to 100cc, the answer is 20 x 100 ÷ 1000 or 2 bottles, and the remainder consisting of 20 lots of 5cc of vinegar can safely be ignored.