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Number and Numeration

Mathematics · WAEC and JAMB · SS2 and SS3

Almost every WASSCE objective paper carries eight to ten questions from this area, and Paper 2 usually opens with one. These are the cheapest marks on the paper, and candidates lose them to careless rounding and misread ratios, not to ignorance.

What you need to know

  • A fraction becomes a decimal by dividing the top by the bottom, and a percentage by multiplying that decimal by 100, so 3/8 = 0.375 = 37.5%. Train yourself to move along that chain in either direction without a calculator.
  • Significant figures are counted from the first non-zero digit, not from the decimal point. In 0.004067 the count starts at the 4, so to 2 significant figures it is 0.0041, while to 2 decimal places the same number is 0.00.
  • When rounding, look only at the digit immediately after the cut-off. In 0.0084567 to 3 significant figures, the digits kept are 8, 4, 5; the next digit is 6, so the 5 becomes 6 and the answer is 0.00846.
  • Standard form is written A x 10^n where A is at least 1 but less than 10. Writing 45.6 x 10^3 is not standard form; it must be 4.56 x 10^4. A number smaller than 1 gives a negative power: 0.00072 = 7.2 x 10^-4.
  • To share an amount in a ratio, add the parts first to get the total number of parts, divide the amount by that total to get the value of one part, then multiply. Never divide the amount by the number of people.
  • A ratio has no units and must be reduced to its lowest terms in the same unit. Comparing 50 kobo to 2 naira gives 50:200 = 1:4, not 50:2.
  • Direct proportion means that as one quantity rises the other rises by the same factor; inverse proportion means one rises as the other falls. For 'if 6 men build a wall in 10 days, how long will 4 men take', the work stays constant, so 6 x 10 = 4 x t and t = 15 days.
  • Percentage change always uses the ORIGINAL value as the denominator. A trader who buys at N62,000 and sells at N71,300 has gained 9,300, and 9300/62000 x 100 = 15%, not 9300/71300.
  • In any number base n, place values are powers of n read from the right: 2134 in base five means 2x5^3 + 1x5^2 + 3x5 + 4 = 250 + 25 + 15 + 4 = 294 in base ten.
  • To convert from base ten to another base, divide repeatedly by the new base and read the remainders UPWARDS. Dividing 45 by 2 repeatedly gives remainders 1,0,1,1,0,1 read from the bottom, so 45 = 101101 in base two.
  • No digit in a base may be equal to or larger than the base, so 182 in base seven is impossible because 8 is not a base-seven digit. Examiners use this to catch candidates in 'find the value of n' questions.
  • When adding or multiplying in a base, carry in groups of that base: in base five, 3 + 4 = 7 which is one group of five remainder 2, so you write 2 and carry 1.
  • Follow BODMAS strictly. In 20 - 4 x 3 + 6/2 the multiplication and division come first, giving 20 - 12 + 3 = 11, not 51.
  • For estimation questions, round every figure to one significant figure before working: 48.7 x 0.0213 / 9.8 becomes 50 x 0.02 / 10 = 0.1, which is close enough to pick the right option.

Key terms

Significant figures
The digits of a number counted from the first non-zero digit onwards, which carry the reliable information in a measurement.
Standard form
A number written as A x 10^n where 1 is less than or equal to A, A is less than 10, and n is an integer.
Ratio
A comparison of two or more quantities of the same kind, expressed without units and reduced to lowest terms.
Inverse proportion
A relationship in which the product of the two quantities stays constant, so one doubles when the other halves.
Percentage error
The error in a measurement expressed as a percentage of the true or actual value.
Number base
The size of the grouping used in a numeral system, which fixes the place values as successive powers of that number.
Rational number
Any number that can be written as a fraction p/q where p and q are integers and q is not zero.

Formulae

  • percentage change = (change / original value) x 100
  • percentage error = (|measured value - actual value| / actual value) x 100
  • standard form: A x 10^n where 1 <= A < 10
  • share of a ratio = (your part / total parts) x total amount
  • value of a numeral in base n = sum of (digit x n^position), counting positions from 0 on the right
  • inverse proportion: x1 * y1 = x2 * y2
  • direct proportion: y1 / x1 = y2 / x2

Worked examples

Share N45,000 among Ada, Bola and Chidi in the ratio 2:3:4.

  1. Add the parts: 2 + 3 + 4 = 9 parts.
  2. Find the value of one part: 45,000 / 9 = N5,000.
  3. Ada gets 2 x 5,000 = N10,000.
  4. Bola gets 3 x 5,000 = N15,000.
  5. Chidi gets 4 x 5,000 = N20,000.
  6. Check: 10,000 + 15,000 + 20,000 = 45,000.

Find the value of n if 24 in base n is equal to 20 in base ten.

  1. Expand the left side using place value: 24 in base n = 2 x n + 4.
  2. Set it equal to the base-ten value: 2n + 4 = 20.
  3. Subtract 4 from both sides: 2n = 16.
  4. Divide by 2: n = 8.
  5. Check the digits are legal: the digits used are 2 and 4, both less than 8, so base 8 is valid.
  6. Verify: 24 in base eight = 2 x 8 + 4 = 20.

Express 0.00084567 correct to three significant figures, and then write your answer in standard form.

  1. Start counting from the first non-zero digit, which is 8. The three significant digits are 8, 4 and 5.
  2. Look at the next digit, 6. Since 6 is 5 or more, round the 5 up to 6.
  3. The value to three significant figures is 0.000846.
  4. To write in standard form, move the decimal point 4 places to the right to get 8.46.
  5. Moving right makes the power negative, so the answer is 8.46 x 10^-4.

The mistake to avoid

Candidates confuse significant figures with decimal places on small numbers. Asked for 0.004067 to 2 significant figures they write 0.00, because they started counting at the decimal point instead of at the first non-zero digit. The second trap is finding percentage profit by dividing by the selling price instead of the cost price.

In the exam

Read whether the question says significant figures, decimal places or nearest whole number, and write the answer in exactly that form, because the final accuracy mark depends on it. Keep full accuracy in your working and round only at the last line. In ratio questions, always add your shares back together as a check before moving on.