N1 - Numbers and Arithmetic

N1 - Numbers and Arithmetic

N1.1 = Rounding

N1.2 = Upper and Lower Bounds

N1.3 = Multiplying and dividing

N1.4 = Estimation

N1.5 = Fraction Calculations

N1.6 = Fractions and Decimals

N1.7 = Fractions, Decimals and Percentages

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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.

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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...

E.G. 

  • 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

 

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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... 

÷ 4 = 9 x ¼ = 2¼ 

  • Multiplying  by a number is the same as dividing by it's reciprocal...

2 x 4 = 2 ÷ ¼ = 8

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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 !

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!

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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

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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


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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%

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