Numeracy 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. Adding Fractions Subtracting Fractions Multiplying Fractions Dividing Fractions (C) Fractions, Decimals and Percentage Calculating a Percentage Finding a Percentage Decimal Places Significant Figures (C) Standard Form Ratio Distance, Speed and Time Simple Interest Compound Interest (C) Depreciation and Appreciation (C) Exchange Rates Direct Relationships Finding the Average (mean) Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 1 1) Adding Fractions Example (Using the "top heavy" method): 1 23 + 2 34 5 + 11 3 4 ~change to "TOP HEAVY" fractions = 20 + 33 12 ~use a COMMON DENOMINATOR = 53 = 4 5 12 12 ~change to a MIXED NUMBER Answer: 2) Subtracting Fractions Example (Using the "top Heavy" method): 3 12 − 1 34 7−7 2 4 ~change to "TOP HEAVY" fractions = 14 − 7 4 ~use a COMMON DENOMINATOR = 7 4 Answer: 3) = 13 4 ~change to a MIXED NUMBER Multiplying Fractions Example: 3 13 % 1 45 10 % 9 3 5 Answer: = 1 = ~change to "TOP HEAVY" fractions 3 2 10 % 9 3 5 6 = 1 ~cancel top and bottom numbers 1 using common factors 6 ~change into mixed numbers. Note: Find out how to use the a b/c button (if you are lucky enough to have one) on your calculator. Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 2 4) (C) Dividing Fractions Example: 2 13 + 5 6 7+5 3 6 = = 5) 7%6 3 5 1 14 5 ~change to "TOP HEAVY " fractions 2 = ~turn last fraction up-side-down 1 PERCENTAGE (multiply by 100) DECIMAL (Divide top by bottom) 2 0.5 50% 4 0.75 75% 8 0.125 12.5% 5 8 0.625 62.5% 7 0.7 70% 1.25 125% 3 1 10 1 14 Calculating a Percentage Example: Find 45% of £300. Answer: 7) ~change to a MIXED NUMBER Fraction, Decimals and Percentage FRACTION 6) and multiply 24 5 0.45 X 300 = £135 ~change the percentage to a decimal Finding a Percentage Example: Out of a class of 30, 5 people are absent. Express the absences as a percentage of the total in the class. Answer: Amount you are interested in Total amount = X 100% 5 % 100% 30 = 16.67% Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 3 8) Decimal Places Rule: Always round up if the next figure is a 5 or greater. Examples: 9) 3.1415927 = 3.1 3.1415927 = 3.14 3.1415927 = 3.142 3.1415927 = 3.1416 3.1415927 = 3.14159 3.1415927 = 3.141593 to 1 decimal place to 2 decimal places to 3 decimal places to 4 decimal places to 5 decimal places to 6 decimal places 4.396 to 2 decimal places = 4.40 (C) Significant Figures The first non-zero digit from the left is the first significant digit. Examples:4890000 8526400 399400 399400 0.00345 0.01009 0.55841 = 4900000 = 8526000 = 400000 = 399000 = 0.0035 = 0.010 = 0.6 to 2 significant figures to 4 significant figures to 2 significant figures to 3 significant figures to 2 significant figures to 2 significant figures to 1 significant figure 10) Standard Form To express a very large or very small number in the form a % 10 n where n is an integer and a is a number between 1 and 10. Examples: = 4.8 % 10 6 = 2.61 % 10 7 = 1.399 % 10 9 = 2.37 % 10 −4 = 5.62 % 10 −7 4,800,000 26,100,000 1,399,000,000 0.000237 0.000000562 When multiplying numbers together in standard form you add the powers. (3.4 % 10 5 )(5.8 % 10 7 ) Example: = 3.4 x 5.8 x 10 5+7 = 19.72 x 10 12 = 1.972 x 10 13 When dividing numbers in standard form you subtract the powers. (3.4 % 10 5 ) + (1.7 % 10 7 ) Example: 3.4 + 1.7 x 10 5−7 = 2.0 % 10 −2 = Find out how to use the EXP key on your scientific calculator. Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 4 11) Ratio Example: Divide the sum of £1000 in the ratio of 3:2. Answer: 3+2=5 1000 + 5 = 200 ~add the numbers in the ratio ~divide the amount by this sum 3 X 200 = £600 and 2 X 200 = £400 ~find answers by multiplying 12 Distance, Speed and Time Example: A train is travelling at 60 km/h. How far will it have travelled after 3 hours 30 minutes? Answer: D = ?, S = 60 km/h, T = 3.5 hours D From the triangle, D = S x T = 60 x 3.5 = 210 km Example: S T Calculate the time it would take to travel 400 miles at a speed of 250 mph (miles per hour). Answer: D = 400 miles, S = 250 mph, T=? D From the triangle, T = D S T= 400 250 S = 1.6 T (0.6 hours is 0.6 x 60 = 36mins) T = 1 hour 36 minutes REMEMBER: Always use units of measurement which are consistent. 13) Simple Interest Example: A man invests £500 in an account which offers 11% simple interest on the amount invested. How much will he have after 5 years? Answer: 11% of £500 = 0.11 X 500 = £55 ~find interest for 1 year 5 X £55 = £275 ~find interest for 5 years Amount in bank after 5 years = principal + interest = £500 + £275 = £775 Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 5 14) Compound Interest (C) Note: Interest is calculated at the end of every year (or perhaps half year) and is calculated on whole number of pounds (pence ignored). Example: Calculate the compound interest made on a principal sum of £500 over 5 years with a fixed interest rate of 11%. Answer: Year Principal Interest(11%) New Amount 1 500 55 555 2 555 61.05 616.05 3 616.05 67.76 683.81 4 683.81 75.22 759.03 5 758.94 83.49 842.52 Total Interest = £842.52 - £500 = £342.52 Note: Compound interest makes more money than simple interest (compare this example with the previous one). 15) Depreciation and Appreciation (C) This is calculated in the same way as compound interest. Example: The value of a car depreciates by 12% each year. If the car is initally worth £5000, how much will it be worth after 3 years? Answer: Year Value Depreciation (12%) New Value 1 5,000 600 4,400 2 4,400 528 3,872 3 3,872 464.64 £3407.36 Example: A valuable gem appreciates in value at the rate of 7% in the first year, 10% in the next and 8% in the third year. If the original value was £2000, how much will it be worth after 3 years? Answer: Year Value % Rate Appreciation New Value 1 2,000 7 140 2,140 2 2,140 10 214 2,354 3 2,354 8 188.32 £2542.32 Note: The rate of interest (or appreciation/depreciation) may vary from year to year (as in the last example). Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 6 16) Exchange Rates Example: Suppose the exchange rate is £1 = $1.66 (U.S. Dollars). Answer: a) How many dollars can you buy for £55.60? Pounds £1 £55.60 Dollars $1.66 $1.66 x 55.6 = $92.30 b) At the same exchange rate, how many pounds could you buy for $200? Dollars $1.66 $1 $200 Pounds £1 £1 1.66 = £0.602 £0.602 x 200 £120.48 17) Direct Relationships Example: If 1 litre of fuel cost £1.36 :Answer: a) How much will 15 litres cost? Litres 1 15 Cost(£) 1.36 1.36 x 15 £20.40 b) How many litres of this fuel could you buy for £30? Cost (£) 1.36 1 30 Litres 1 1 1.36 = 0.735 0.735 x 30 22.1 Litres 18) Finding the Average (mean) An average is a typical value from a set of values. Example: Find the average of 8, 10, 15, 40, 56 and 111. Answer: Average (mean) = 8 + 10 + 15 + 40 + 56 + 111 = 40 6 Knightswood©Copyright Kayar publishers 2000 Outline Course Notes Numeracy page 7