Rounding - Grade D
Numbers are rounded to make them easier to handle...
- Numbers are rounded up if the 'next' digit is 5 or more, e.g.
8374 rounded to the nearest 10 = 8370
- When rounding to a given number of significant figures, start counting at the first non-zero digit, e.g.
16,487,593 rounded to 2 SF (significant figures) = 16,000,000
- Round measurements to a realistic degree of accuracy, e.g.
If you are using a meter rules, give your answer to the nearest half centimeter.
If the next digit is 5 or more, round up.
For SF, start counting at the non-zero digit.
Upper and Lower Bounds - Grade B/A
If the height of a tree is given as five meters, correct to the nearest meter...
It could be anywhere between 4.5 meters and 5.5 meters...
4.5m = lower bound
5.5m = upper bound
- The upper bound is not actually included in the range of possible values...
- The lower bound is the lowest measurement a rounded value could be...
- The upper bound is the highest measurement a rounded value could be...
- The upper and lower bounds of 5.8m = 5.85m/5.75m
- The upper and lower bounds of 35mm, correct to the nearest 5mm = 37.5mm/32.5mm
Multiplying and Diving - Grade D
Multiplying doesn't always make numbers bigger, and dividing doesn't always make numbers smaller...
- Multipying by numbers less than 1 = makes positive numbers smaller
- Dividing by numbers less than 1 = makes positive numbers bigger
- Multiplying/Dividing by 1 = no change
- Multiplying by numbers more than 1 = makes positive numbers bigger
- Dividing by numbers more than 1 = makes positive numbers smaller
E.G. 8 x 0.5 = 4
- Dividing by a number is the same as multiplying by it's reciprocal...
9 ÷ 4 = 9 x ¼ = 2¼
- Multiplying by a number is the same as dividing by it's reciprocal...
2 x 4 = 2 ÷ ¼ = 8
Estimation - Grade C/B
You can use approximations to one SF to make estimates...
You need to be careful when estimating with powers:
- 1.3 is quite close to 1, but 1.3³ is not close to 1³!
E.G. (563 + 1.58) ÷ (327 - 4.72) = 600 ÷ 300 = 200
200 = estimate
- You can use standard form to make estimations with very large or small numbers
Estimating involves rounding any numbers that are hard to work with!
Fraction Calculations - Grade D/C
You can add or subtract fractions if they have the same denominator...
- You can make two demoninators the same by multiplying them together, but whatever you do to the denominator, you have to do to the numerator, too.
½ + ¼ = 4/8 + 2/8 = 6/8 = 3/4
- To multiply fractions, multiply the numerators together, and multiple the demoninators together.
¼ x ½ = 1/8
- Dividing by any number is the same as multiplying by its multiplicative inverse (e.g. 4/1 is the multiplicative inverse for 1/4).
½ ÷ ¼ = ½ x 4/1 = 4/2 = 2/1 = 2
Fractions and Decimals - Grade B/A
By dividing the numerator by the denominator, you can convert a fraction to a decimal!
Terminating or Recurring?
Terminating = the denominator has only got the factors 2 and 5 (e.g. 8= 2x2x2)
Recurring = the denominator has got factors other than 2 and/or 5 (e.g. 6 = 2x3)
To convert a terminating decimal to a fraction, write the decimal as the numerator (e.g. 0.385 as 385) and the denominator as a power of 10.
0.385 = 385/1000 = 77/200
To convert a recurring decimal, do the following...
x = 0.36(recurring) > 100x = 36.36(recurring) > 99x = 36 > x = 36/99 = 4/11
Fractions, Decimals and Percentages - Grade C/B
Converting between Fractions, Percentages and Decimals...
- To convert a decimal to a percentage, multiply it by 100%.
0.65 = 0.65 x 100% = 65%
- To convert a percentage to a decimal, divide it by 100.
18.3% = 18.3 ÷ 100 = 0.183
- To convert a fraction to a percentage, divide the numerator by the denominator and then multiply it by 100%.
5/8 = 5÷8 = 0.625 x 100% = 62.5%