Measurement!(MM1)! Units!of!Measurement!and...

1 General Mathematics (Preliminary Course) | Units of Measurement and Applications

Measurement (MM1)

Units of Measurement and

Applications

Name ...................................................................................................

G.Georgiou


• Repeat and Average Measurements to Reduce the Likelihood of Error Measure the length of the line below correct to the nearest millimetre. Record your answer next to it. As you may have noticed, some students in your class or around you have achieved a different answer to you. This is common with measuring instruments. Often the measurement you take can vary, from measurement to measurement. Think about the speedometer on your parent’s car. It can often be very difficult to take an accurate measurement as the car is travelling. One strategy to alleviate this problem is to take several measurements and then average your results. Carl the carpenter was installing a window in a house, and had to make a hole through some gyprock. To determine the length of the hole he measured the length of the window three times correct to the nearest millimetre. He obtained the following measurements:

1112 mm, 1114 mm, 1109 mm

a) What should Carl do to ensure he obtains an approximately correct measurement? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... b) What measurement should Carl use? .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 1


• Investigate the Degree of Accuracy of Reported Measurements, including the Use of Significant Figures where Appropriate

Thus far, the main form of rounding used has been rounding to a given number of decimal places. This requires you to examine the number after the amount of decimal places you need and determining whether to round up or stay the same. Round the following number correct to 3 decimal places. (a) 2.4852 ........................................................................................ (b) 3.2419456 ........................................................................................ (c) 3.2987 ........................................................................................ (d) 2.89954 ........................................................................................ Imagine someone measured the distance between the Earth and the Sun as 149 496 347 km. Unless you were certain about this measurement, it is quite difficult to quote it as accurate. Hence, it is more appropriate to estimate this distance as 150 000 000 km (correct to 2 significant figures) or 149 000 000 km (correct to 3 significant figures). When rounding with significant figures, you start counting from the first non-‐zero digit. It is a method of rounding that works accurately both with decimals and whole numbers. Round the following numbers correct to 3 significant figures. (a) 325 422 ........................................................................................ (b) 0.01485 ........................................................................................ (c) 0.000 002 344 ........................................................................................ (d) 388 724 ........................................................................................ (e) 2 972 000 ........................................................................................ Note: while zeros are mostlyk insignificant (unless between other non-‐zero digits), they are often essential for place value.

Example 2

Example 3


Calculate the following and round your answer off to 2 significant figures.

(a) !"#

$"#+#"% (b) !"#$

%! $!

.............................................................. ........................................................................ .............................................................. ........................................................................ .............................................................. ........................................................................ A farmer measured the length of her paddock to be 50 m long. What is the largest and smallest possible length the paddock could be, given the farmer rounded her answer to 2 significant figures? Would we have a different answer if the measurements were rounded correct to 1 significant figure? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 4

Example 5


Jane measured the length of a piece of paper to be 29.23456cm, using a ruler marked in cm. (a) Is this a reasonable measurement? Explain your answer. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (b) To how many significant figures would you need to round this number to make the measurement more accurate? .................................................................................................................................................... .................................................................................................................................................... Which of the following measurements is more accurate, 38.2 cm or 38.20 cm? Explain your answer. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 6

Activity Ex 6:02 Q 6, 12 Ex 6.03 Q 1, 2, 3, 4, 5, 7, 8, 9

Example 7


• Use of Positive and Negative Powers of Ten in Expressing Numbers in Scientific Notation

Scientific notation provides us with an easier way of writing very large or small numbers. When using scientific notation all digits can be of the form ! !!"" , where n is a whole number and m is a number between 1 and 10. Write the following numbers in scientific notation.

(a) 23 400 000 (b) 7 000 000 000

.............................................................. ...................................................................... .............................................................. ...................................................................... (c) 72 010 000 (d) 0.00023 .............................................................. ...................................................................... .............................................................. ...................................................................... (e) 0.000 003 02 (f) 0.000 000 1 .............................................................. ...................................................................... .............................................................. ...................................................................... Convert these numbers from scientific notation to normal notation.

(a) 3.2 × 10 5 (b) 3 × 10 4

.............................................................. ...................................................................... .............................................................. ...................................................................... (c) 3.1 × 10 – 4 (d) 3 × 10 – 1 .............................................................. ...................................................................... .............................................................. ......................................................................

Example 8

Note: the power is positive for numbers larger than 1 and negative for numbers less than 1

Example 9


Evaluate the following: (a) (3.2 × 104) × (2.1 ×10-‐3) (b) (3.4 × 105) – (4.2 ×10-‐1) .............................................................. ...................................................................... What is the correct way of writing the number 325 000 000 with two significant figures?

(A) 330 (C) 3.25 x 102 (B) 3.2 x 108 (D) 3.3 x 108

Example 10

Example 11

Example 12


.................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

a) Explain why 33.7 × 105 is not written in scientific notation. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

b) Hence, write 33.7 × 105 in scientific notation. ..................................................................................................................................................................

Activity Ex 6.04 Q 1, 2, 4, 6, 8

H.S.C. Question (14)

Example 13


• Calculate with Ratios, including Finding the Ratio of Two Quantities, Dividing Quantities into a Given ratio and Using the Unitary Method to Solve Problems

Ratios compare two values that are in the same units. An example of a ratio is when mixing some sort of detergent. The bottle may say 1 part detergent to 2 parts water. In Mathematics we write 1:2. It is important to remember the following:

• Ratios are always expressed as whole numbers in simplified form (unless otherwise stated).

• Ratios are in the same units. • The final answer in a ratio should never have units attached.

Simplify the following ratios: (a) 5 : 10 (b) 15 : 10 (c) ½ : ¾ ..................................... ................................................ ..................................................... (d) 5ab : 15a (e) 1.2 : 3.4 (f) 3.5 : 5 ..................................... ................................................ ..................................................... (g) 5 mins : 10 secs (h) 12 : 36 : 9 (i) 2.5: 0.5 : 0.25 ..................................... ................................................ ..................................................... ..................................... ................................................ ..................................................... One of the techniques used to solve ratio problems involves dividing the quantity into a given ratio. Ruth and Peter invested $50,000 in the ratio 3:2 respectively. Calculate the amount they both invested. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 15

Example 16


Concrete is composed of sand, cement, and limestone. These components are required in the ratio 4 : 5 : 2. If 22 kg of concrete is required, how much of each component is required? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... Another technique used to solve ratio problems is the unitary method. Evelyn bought a commercial shampoo cleaner. The instructions state to dilute the cleaner with water in the ratio 1 : 25. If Evelyn uses 75 mL of water, how much cleaner does she require? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... Dates, flour and sugar are mixed in the ratio 5 : 6 : 1 to make date pudding. How much flour and sugar are required if 300 grams of dates are used? .................................................................................................................................................... .................................................................................................................................................... ................................................................................................................................................... ....................................................................................................................................................

Example 17

Example 18

Example 19


.................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 20


.................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

Example 21

Activity Ex 6.05 Q 1, 2, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17


• Calculation of Rates (Eg. pay rates, speeds, rates of flow) A rate compares two different quantities. Examples of rates include:

• ....................................................................

• ...................................................................

• ...................................................................

We can calculate rates quite easily if we are given some basic initial data. A car travels 400 km in 5 hours. What was its average speed in km / hr? .................................................................................................................................................................. .................................................................................................................................................................. A typist typed a 1500 word essay in 30 minutes. What was the average typing speed in words / minute? .................................................................................................................................................................. .................................................................................................................................................................. It takes an athlete 10 seconds to run 100 m. Calculate the average running speed in m / sec. .................................................................................................................................................................. .................................................................................................................................................................. Petrol costs $1.35 per litre. Calculate how much it would cost to fill a tank with 40 litres of petrol. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

Example 22

Example 23

Example 24

Example 25


A plane flies a distance of 4127km at a speed of 940km/h. How long will it take to get to its destination? .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................


Example 26


....................................................................................................................................... ....................................................................................................................................... .......................................................................................................................................




....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... ....................................................................................................................................... .......................................................................................................................................

Activity Ex 6.07 Q 3-‐14


• Conversion between Units for Rates (Eg. km / h to m / s) If an athlete can run 100m in 10 seconds, then imagine how far he could run in an hour if he could maintain that speed. Convert 84 beats / min to beats / sec. .................................................................................................................................................................. .................................................................................................................................................................. Convert 60km / hr to m / sec. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. Convert 100 m / 10 seconds to km / hr. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. If the reaction time of a car driver is 0.9 seconds, how far will the car travel if its speed is 60 km / hr? .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

Example 31

Example 32

Example 33

Activity Ex 6.08 Q 1, 2, 3, 5, 7, 8, 10

Example 30


• Increasing and Decreasing an Amount by a Percentage Stores often reduce or increase their prices for a variety of reasons. Solve the following problems. (a) Increase $140 by 7.5%. .................................................................................................................................................... .................................................................................................................................................... (b) At a store sale there was 15% off. If a pair of shoes cost $129, calculate the discounted price. .................................................................................................................................................... .................................................................................................................................................... • Determination of Overall Change in a Quantity Following Repeated Percentage

Changes A common misconception is that a discount of 10% and then a discount of 5% is equal to an overall discount of 15%. However this belief is incorrect. At a store a product was $550. The salesperson offered the customer a discount of 10%. However when the salesperson realised the customer was a Gold customer, he offered him a further 5% discount. (a) Calculate the price of the product after the 10% discount. .................................................................................................................................................... .................................................................................................................................................... (b) Calculate the price of the product after a further 5% was deducted from it. .................................................................................................................................................... ....................................................................................................................................................

Example 34

Example 35

Recall


(c) Calculate the price of the product if 15% was deducted from the original amount. .................................................................................................................................................... .................................................................................................................................................... (d) The salesperson believes that a discount of 15% is the same as discounting a product by 10% and then 5%. Using your answers from the questions above, justify why the salesperson is incorrect. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... This now leaves the question: what is a discount of 10% followed by 5% equal to. We can determine this by multiplying the two percentages in their increased/decreased form. (a) What single percentage discount is equivalent to giving a discount of 10% then 5%? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (b) What percentage change is equivalent to an increase of 10% then an increase of 14%? .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (c) What percentage change is equivalent to a decrease of 8% then an increase of 12%? .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Example 36

Activity Ex 6.09 Q 1, 2, 3, 7, 8, 9, 10, 11


• Convert between Common Units for Area and Volume

Units are essential in any type of measurement. Length Provide some examples of items that are measured in these units. .................................................................................................................................................... Area Write down some examples of items that are measured with these units. .................................................................................................................................................... This is a square mm. This is a square cm. Volume Write down some objects that might be measured using these units. ....................................................................................................................................................


Convert the following into the required form.

Question Working Space

1005 m = ______________km

25 mm = _____________ cm

55 cm2 = ______________mm2

3 000 000 cm3 = _______________ m3

24 000 m2 = ______________ha

3 km2 = __________________m2

Ashley was asked to convert the area of her garden that was 22 m2 to cm2. She simply divided by 100 because she stated there were 100 cm in 1 metre. Ashley has made two errors, explain what they are. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

Example 37

Example 38

Activity Ex 6.01 Q 1 Ex 6.11 Q 1,2,4,5 Ex 6.13 Q 1


Literacy Question 1 Explain the difference between a ratio and a rate. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. Question 2 Explain why scientific notation is used. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. Question 3 Explain why rounding to significant figures is more advantageous than rounding to decimal places. .................................................................................................................................................................. .................................................................................................................................................................. .................................................................................................................................................................. ..................................................................................................................................................................

Measurement!(MM1)! Units!of!Measurement!and...

Documents

Transcript of Measurement!(MM1)! Units!of!Measurement!and...