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1 BIRKBECK MATHS SUPPORT Measurements 1 In this section we will look at - Examples of everyday measurement - Some units we use to take measurements - Symbols for units and converting between units - Using powers or indices in units - Converting units with powers or indices - Combinations of units and how to convert these Helping you practice At the end of the sheet there are some questions for you to practice. Don t worry if you can t do these but do try to think about them. This practice should help you improve. I find I often make mistakes the first few times I practice, but after a while I understand better. Online Quizzes and Videos There are online quizzes and videos covering examples related to this worksheet. All the answers to questions in this worksheet are available at the website Good luck and enjoy!

2 1. Measurements Measurements and units are important in everyday life as well as in health, engineering and science related jobs. You may need to understand the difference between ounces and grammes for cooking, kilometres and miles for driving or pounds and dollars when travelling overseas. For some jobs understanding measurements and units is a matter of life or death. If a patient needs 50 milligrammes of a drug and the solution has 250 grammes per litre, how can we calculate how much the patient should drink? Here we will show how to make measurements and calculations in different units. Once we ve learnt how to convert one type of unit, the same rules apply for all units. As usual we will start with examples of every day measurements to show how the maths works then we ll try some harder examples using the same ideas. Here are some everyday examples of measurements and units: The news is on in 10 minutes The park is about 3 miles away We need 2 pints of milk Some chocolate bars contain more than 20 grammes of fat The chicken must be cooked at 400 degrees Fahrenheit For each of these examples the measurements are the numbers and the unit shows what you are measuring. So minutes is a unit for measuring time, miles is a unit for measuring distance, pints is a unit for measures volume, grammes is a unit for measuring mass (which is sometime called weight) and finally degrees Fahrenheit is a unit for measuring temperature. But for each of these measurements we could have used a different unit. For example we can measure time using the units of hours or weeks. We can measure distance using the units of metres or miles. We can measure volume using the units of litres instead of pints, we can measure mass using ounces and we can measure temperature using degrees Celsius instead of Fahrenheit.

3 So the first thing we will look at is how to convert between different units for the same measurements, here is a quick example If a caterpillar is 2 inches long, can we convert this to metres? 1 inch is about 2.5 centimetres so we would write this as 1 inch = 2.5 centimetres So 2 inches is the same as 2 x 2.5 centimetres = 5 centimetres...and 3 inches = 3 x 2.5 centimetres = 7.5 centimetres...and 10 inches = 10 x 2.5 centimetres = 25 centimetres But 'centi' means one hundredth or 0.01, so 1 centimetre = 0.01 metre. This means 5 centimetres is 5 x 0.01 metre = 0.05 metres So if we think about our original caterpillar, which is 2 inches long, we can say 2 inches = 2 x 2.5 centimetres = 5 centimetres = 5 x 0.01 metres = 0.05 metres Don t worry if you didn t know that 1 inch is 2.5 centimetres or that 1 centimetre is 0.01 metre, we will cover all these things and more below. But before we move on we will quickly introduce some of the important symbols for units SOME SYMBOLS FOR UNITS Here are some common units and their symbols Unit Symbol Measurement Metre m length or distance Second s time Kilogramme kg mass (often called weight) As an example we would write 5 metres as 5m. We would write 30 seconds as 30s and we would write 11 kilogrammes as 11kg.

4 SOME SYMBOLS FOR PREFIXES A prefix is something that goes in front of a unit. In the word kilometre the kilo part is the prefix and the metre part is the unit. There are lots of prefixes such as centi, kilo, milli. Here are the most common Prefix Symbol Amount micro µ milli m centi c deci d kilo k Mega M Giga G The symbol for micro is one of the Greek letters of the alphabet,. This letter is called mu which rhymes with phew and is pronounced a bit like myou. Combining units and prefixes It is common to combine prefixes and units, for example, from the two tables above we can see that a centimetre is made up of the prefix centi and the unit meter, so together we would write this as cm which means 0.01 meter. So 7 millimetres = 7 mm = 7 x x metre = metres = 0.007m Combining units together You will often see the word per in combinations of units, for example kilometres per hour, grammes per litre. The word per is equivalent to dividing. Here we have written m/s and g/l in three different ways, with a divide sign ( ), as a fraction and with indices. Each of these ways means the same thing (metres per second) and the measurements are calculating distance divided by time (for m/s) and mass divided by volume (for g/l). First though, we will look at how to convert between different units that measure the same quantity.

5 2. Introduction to converting units We start by converting units of time, length, mass and temperature. CONVERTING UNITS FOR TIME We know that there are many ways to measure time; we can go on holiday for a fortnight which is the same as 2 weeks or fourteen days. So we can write this mathematically as 1 fortnight = 2 weeks = 14 days But other units for time include seconds, years, months, hours and we can also write equations to relate these 1 year = 365 days 1 day = 24 hours = 24 x 60 minutes = 24 x 60 x 60 seconds So if a plane journey takes 8 hours, this is the same as 8 hours = 8 x 60 minutes = 8 x 60 x 60 seconds = 28,800 seconds Or if we go on holiday for a fortnight, we can work how many hours this is, as 1 fortnight = 14 days = 14 x 24 hours = 336 hours CONVERTING UNITS FOR LENGTH There are lots of units for length, for example A wall is two meters high The plane is flying at 37,000 feet He has a five centimetre scar on his arm Newcastle is about 200 miles from Dublin Viruses are about one tenth of a micrometre across And again we can convert between all these measurements as long as we know the conversion rate between the different units.

6 Example 1: If we want to convert 10 miles to metres and we know that 1 mile is 1.6 km, we first need to convert miles to kilometres, and then convert kilometres to metres. So we start with 1 mile = 1.6 km, therefore But kilo means 1,000, therefore Or writing this in scientific notation (Scientific notation and indices are covered in the Numbers and Calculations worksheets on Example 2: To convert 50 kilometres into miles using the fact that 1 mile is 1.6 km then we have a different problem. We still have to start with 1 mile = 1.6 km but this converts from miles km (as in Example 1) and not from km miles (as needed in this example). So we need to find a way to write 1 km in terms of miles. So we start by swopping sides of the previous equation and we write If we now divide both sides by 1.6 we get The left-hand side just gives us 1km (because 1.6 divided by 1.6 is 1), so Where we used a calculator to re-write. We can now calculate

7 CONVERTING UNITS FOR MASS It doesn t matter which units we are converting as the rules are always the same. But we do need to be a little careful when using the word mass. Most of the time people use the word weight to mean mass, for example we might say 'the weight of a bag of sugar is about 1 kilogramme'. However some scientists use weight to mean the force. In fact weight is the force that a mass exerts due to gravity (so the weight is the force you would feel if someone drops a bag of sugar on your toe). For this reason we will use the word mass instead of weight. There are many units for mass (e.g tonnes, stones, grammes or ounces) and if we know what each unit is in terms of any of the others we can always convert. Example: If 1 ounce is 28 g, what is 320 grammes in ounces? Well we have But this only tells us the conversion of ounces grammes and we need to go the other way around, ie grammes ounces, so we need to write CONVERTING UNITS FOR TEMPERATURE For cooking and when describing the temperature as part of the weather we use Celsius or sometimes Fahrenheit (F). There is another scale for temperature that engineers or scientists will regularly use called Kelvin (K). If the temperature is colder than we use negative numbers, so means 5 degrees below zero. The coldest temperature possible Absolute Zero is, and this is the start of the Kelvin scale. So. In general To convert between Fahrenheit and degrees Celsius we use the formula below, and we will consider this further in the algebra section.

8 3. Units with powers: area and volume We start by thinking about the meaning of area. One way to do this is to consider how much carpet we need to cover a floor. Suppose we have a room that is 5m long and 3m wide and we are fitting carpet tiles that are each 1m long and 1m wide, we can draw the room as 3m 5m We can count or calculate the number of tiles needed which is 3 x 5 = 15 This calculation relates to measuring the area of a rectangle and the formula for area of a rectangle is But here we are particularly interested in the units. The units we use to measure length are metres [m] and the units we use to measure the width are also metres [m]. So to find the units of area, we write So in this case the area is. But we could have measured the length and width in any units measuring distance (for example feet and inches or centimetres) and the units we use will affect the results.

9 So let s measure the same area in centimetres. The length 5m is 500cm and the width of 3m is the same as 300cm, so the diagram of the same room is now 300 cm 500 cm Using the formula for area of rectangle from above we get Notice how the unit is now when previously it was. Also notice and that the area originally calculated as is the same as. One way to think about this is to notice that it would take 15 tiles of size 1m by 1m to cover the floor, but 150,000 tiles of size 1cm by 1cm to cover the same area. By comparing these two ways of measuring the area (the first one used metres squared and the second one used centimetres square) we now know that But rather than drawing out and counting floor tiles every time we want to convert units, we can use the rules of indices. We start with:,, So 1 square centimetre, which is 1cm x 1cm is written as Here we have used the rules of indices: if a power is raised to a power we multiply the powers (in this case -2 x 2 = -4). So we can now see that if

10 METERS SQUARED TO FEET SQUARED Let s now convert a measurement of area from metres squared ( ) to feet squared ( ). We start with the conversion rate of 1 meter = 3.3 feet and draw our diagram again but with the length and the width in units of feet. 3m = 3 x 3.3 feet = 9.9 feet 5m = 5 x 3.3 feet = 16.5 feet Using the width as 9.9 feet and the length as 16.5 feet, we can now calculate the area from the formula to give an answer in units of But we could arrived at the same answer if we started with So now we can use our original to write So we get exactly the same answer and it doesn t matter if we draw the diagram or just convert the unit. However we must be careful and make sure that if the unit is squared then also square the prefix and the conversion. Key Point: When converting between units with powers we must always remember that the powers applying to the units also automatically apply to the conversion and the prefix.

11 Example 1: The International Football Association has set the size of a football pitch to be 105m long and 68m wide. Calculate the area in meters squared then using, convert this into feet squared. We start by writing out the information we know, then we calculate the area: Now we want to convert into, first we calculate in So we know that is the same as and we can convert the area of the football pitch (which is ) into using We can double check this answer by calculating the length and width in feet: So we can then calculate the area directly in units of feet squared, Example 2: One tonne of Carbon Dioxide,, at standard pressure and temperature occupies a volume of. Convert this to.

12 VOLUME Having spent some time on units for areas we will now consider volume. The volume of an object measures how much space it takes up. Some every day examples of units for volume include pints and litres. There are lots of situations when it is important to understand or convert a measurement for volume. A recipe may need of water, or a patient may need of blood or a chemical reaction may need of acid. The volume of an object can depend on its shape, and we start with a box with length 10m, height 5m and width 2m. height = 5m width = 2m length = 10m The volume of a box is calculated by multiplying the length, width and height. With any measurement we must always think about units. To measure length, width and height we use the units meters. So the units for volume are And in this case we can calculate the volume as So we can readily see the unit for volume in this case is or metres cubed

13 LITRES Litres are often used to measure volume and a litre is the volume of a box which has each side measuring 10cm 10cm 1 Litre 10cm 10cm CONVERTING BETWEEN LITRES AND METRES CUBED To convert between litres and we first notice that 10cm is the same as 0.1m. This means our volume of 1 litre, which is a box with all sides measuring 10cm, can also be drawn with all sides 0.1m. This means: 0.1m 1 Litre 0.1m But we also notice that which is one decimetre (dm). This means we can also write 0.1m So 1 litre is the same as 1 decimetre cubed. Examples So if we know we can calculate the following: Example 1: What is converted to? Answer: starting with we can see that

14 Example 2: What is 75ml converted to? Answer: the measurement is given in ml (millilitres) so we first need to convert millilitres to litres. Now we can use, and carry on from the line above to write Example 3: What is converted to? Answer: This question asks us to convert from to. All the other questions so far have asked for to and to do these conversions we used the expression so we need to find a way to write 1 meter cubed in litres, so this is where we start Dividing both sides by gives us On the right-hand side we can now cancel the on the top and the bottom of the fraction to get 1 and on the left-hand side we can move the denominator to the numerator to get, so we can write This is now our starting point for converting into litres, and we get Example 4: What is converted to millilitres ( )?

15 4. Converting units with powers and prefixes We have seen that 1 litre can be written in many ways, Notice here that we have changed between into into, and that these do not change in the same way as into and into. For example Where as This is because when the unit is raised to a power (such as meters cubed, ) the power applies to the pre-fix as well as the unit. Here is an example Remember the key point: When converting units with powers, the powers applying to the units also apply to the prefixes and the conversion factor. Examples: For these examples you should only need the prefix table on page 4, but if you want to revise indices and powers, see the Numbers page of the website Example 1: Convert Answer: we start by using so we can write as If we now want to write this in scientific notation, we write

16 Example 2: Convert Answer: here we need to use but we are converting from meters to millimetres, so we need an expression for meters in terms of millimetres If we now divide both sides by we get so we can write an expression for meters in terms of millimetres Now with this as our starting point we can convert in the following way Which is the answer and it is already in scientific notation. Example 3: Convert Starting with into we can write Where we have shown all the steps in one line and given our answer in scientific notation, Example 4: Convert into Starting with we can write Example 5: Convert into Starting with

17 5. Introducing combinations of Units So far we have considered units such as meters (for distance), seconds (for time) and grammes (for mass). We have also considered the meaning of units with prefixes, such centimetres ( ) or milligrammes ( ) and explored units with powers (volume is measure in and area is measured in ). We have also considered other ways to take measurements, such as using litres to measure volume ( ) or Fahrenheit or Kelvin to measure temperature. There are lots more examples with converting units in the later worksheets and we shall try exercises at the end of this worksheets. But before we finish we will look at combinations of units. SPEED A good example of a using a combination of units occurs when measuring speed. If you drive a car you measure your speed in miles per hour. This unit miles per hour measures the distance [miles] per time [hours], so So the word per means divided by and more generally we can write But we know that distance can also be measured in meters, and time can also be measured in seconds, so from the above formula we can also measure speed in or or. We will explore these ideas in the next worksheet. DENSITY Another common measurement with a combination of units is density. The unit for density is kilogramme per meter cubed or and the formula is We will look at how to convert measurements when we have a combination of units (such as speed or density) in the next section.

18 Now your turn Generally the more maths you practice the easier it gets. If you make mistakes don t worry. I generally find that if I make lots of mistakes I understand the subject better when I have finished. If you want to see videos explaining these ideas and showing the answers visit A) State whether the following are measuring volume, mass, time, temperature or distance 1) 40 milligrams of vitamin E 2) 75g of flour 3) A prescription for 25ml 4) 147 milliseconds 5) 196 Kelvin 6) A journey of 14 miles 7) A journey of 37 minutes 8) A rubbish skip sized 8 cubic yards B) Convert the units with pre-fixes listed below, you may want to use the table in section 1 1) 50 g into kg 2) 5 hours into minutes 3) 0.4 seconds into milliseconds 4) 5 hours into minutes 5) 120 ml into litres 6) 0.72 kg into g 7) 120cm into m 8) 2.53m into mm C) Convert the units listed below, using the information: 1 kilometre = 0.62 miles, 1 ounce = 28 gramme, 1 litre = 1) 14 kilometers miles 2) 15 grammes ounces 3) 2 litres metres cubed 4) 3 inches metres, 1 inch = 2.5 cm 5) 2.1 miles kilometres 6) 7 ounces grammes 7) 4 metres cubed litres 8) 21 cm inches

19 D) Converting between units with different prefixes 1) 2) 3) 4) 5) 6) 7) 8) E) Converting between measurements using units with powers 1) 2) 3) 4) 5) 6) 7) 8) F) Converting between measurements using different units, with prefixes and powers in the units, using information: 1 inch = 2.5 cm, 1 metre = 3.3 feet 1) 2) 3) 4) 5) 6) 7) 8) G) General word problems involving units. 1) The area of Singapore is. If 1 mile is 1.6 km, what is the area of Singapore in square miles? 1) A typical jet engine takes in 1.2 tons of air per second during takeoff. If a ton is 40 cubic feet and 1 metre is 3.3 feet. What is the volume of air per second in metres cubed per second?

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