IGCSE Mathematics: Number — Fractions, Percentages, Standard Form and Estimation

TL;DR

You'll learn how to work confidently with fractions, percentages, and numbers in standard form, which are essential for many IGCSE math problems. We'll also cover estimation, a key skill for quickly checking your answers. Mastering these number concepts will build a strong foundation for the rest of your math studies.

1. The Mental Model

Think of numbers as having different "forms" or "representations" that are useful for different situations. Fractions show parts of a whole, percentages show parts per hundred, and standard form handles very large or very small numbers efficiently. Estimation is your quick "sanity check" to see if your answer makes sense.

2. The Core Material

Fractions

Fractions represent parts of a whole, written as numerator/denominator.

• Adding/Subtracting: You need a common denominator. Find the lowest common multiple (LCM) of the denominators, then adjust the numerators.
  • Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
• Multiplying: Multiply numerators together and denominators together. Simplify before or after.
  • Example: 1/2 * 3/4 = (1*3)/(2*4) = 3/8
• Dividing: "Flip" the second fraction (find its reciprocal) and then multiply.
  • Example: 1/2 ÷ 3/4 = 1/2 * 4/3 = 4/6 = 2/3
• Mixed Numbers to Improper Fractions: Multiply the whole number by the denominator, add the numerator, and put it over the original denominator.
  • Example: 2 1/3 = (2*3 + 1)/3 = 7/3
• Improper Fractions to Mixed Numbers: Divide the numerator by the denominator. The quotient is the whole number, and the remainder is the new numerator over the original denominator.
  • Example: 7/3 = 2 with remainder 1, so 2 1/3

Percentages

Percentages are fractions out of 100. The symbol % means /100.

• Fraction to Percentage: Multiply by 100.
  • Example: 3/4 = (3/4) * 100% = 75%
• Decimal to Percentage: Multiply by 100.
  • Example: 0.25 = 0.25 * 100% = 25%
• Percentage to Decimal: Divide by 100.
  • Example: 75% = 75/100 = 0.75
• Finding a Percentage of an Amount: Convert the percentage to a decimal or fraction and multiply.
  • Example: 20% of 50 = 0.20 * 50 = 10
• Percentage Increase/Decrease:
  • Increase: Original Amount * (1 + (Percentage Increase/100))
  • Decrease: Original Amount * (1 - (Percentage Decrease/100))
  • Alternatively: Find the percentage amount, then add/subtract.
• Reverse Percentages: If an amount after a percentage change is given, you need to find the original amount.
  • If x is the original amount and it increased by 10% to 110, then x * 1.10 = 110. So x = 110 / 1.10.
  • If x is the original amount and it decreased by 10% to 90, then x * 0.90 = 90. So x = 90 / 0.90.

Standard Form (Scientific Notation)

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Used for very large or very small numbers. It's written as A x 10^n, where 1 <= A < 10 and n is an integer.

• Writing in Standard Form:
  1. Move the decimal point until you have a number A between 1 and 10 (inclusive of 1).
  2. Count how many places you moved the decimal. This is n.
  3. If you moved the decimal left, n is positive. If you moved it right, n is negative.
     * Example: 345000 = 3.45 x 10^5 (moved left 5 places)
     * Example: 0.0000078 = 7.8 x 10^-6 (moved right 6 places)
• Converting from Standard Form:
  • If n is positive, move the decimal n places to the right (add zeros).
  • If n is negative, move the decimal n places to the left (add zeros).
  • Example: 2.1 x 10^4 = 21000
  • Example: 5.6 x 10^-3 = 0.0056
• Calculations: When multiplying/dividing, operate on A values and 10^n values separately.
  • (A x 10^m) * (B x 10^n) = (A * B) x 10^(m+n)
  • (A x 10^m) / (B x 10^n) = (A / B) x 10^(m-n)
  • Remember to adjust A and n if A isn't between 1 and 10 after calculation.

Estimation

Rounding numbers to one significant figure (1 s.f.) before calculating to get an approximate answer. This is great for checking if your calculator answer is reasonable.

• Rounding to 1 s.f.: Look at the first non-zero digit. If the next digit is 5 or more, round up. Otherwise, keep it the same. Fill remaining places with zeros.
  • Example: 345 -> 300
  • Example: 0.0789 -> 0.08
  • Example: 7.56 -> 8
• Applying to calculations:
  • Example: (29.7 * 4.1) / 0.098
  • Estimate: (30 * 4) / 0.1 = 120 / 0.1 = 1200

graph TD
    A["Number Concepts"] --> B["Fractions"]
    A --> C["Percentages"]
    A --> D["Standard Form"]
    A --> E["Estimation"]

    B --> B1["Adding/Subtracting (Common Denom)"]
    B --> B2["Multiplying (Numerator*Numerator)"]
    B --> B3["Dividing (Flip & Multiply)"]
    B --> B4["Mixed/Improper Conversions"]

    C --> C1["Fraction/Decimal <-> Percentage"]
    C --> C2["% of an Amount"]
    C --> C3["% Increase/Decrease"]
    C --> C4["Reverse Percentages"]

    D --> D1["Converting to A x 10^n"]
    D --> D2["Converting from A x 10^n"]
    D --> D3["Multiplication/Division Rules"]

    E --> E1["Rounding to 1 s.f."]
    E --> E2["Applying to Calculations"]

3. Worked Example

Let's calculate the final price of an item that originally cost £120, was increased by 15%, and then 1/4 of the new price was paid as a deposit, with the remaining amount being paid by 5 equal monthly installments. Give your final answer in standard form rounded to 2 significant figures.

1. Price after 15% increase:
   £120 * (1 + 15/100) = £120 * 1.15 = £138

2. Deposit paid (1/4 of new price):
   1/4 * £138 = £34.50

3. Remaining amount to pay:
   £138 - £34.50 = £103.50

4. Each monthly installment (5 equal parts):
   £103.50 / 5 = £20.70

5. Convert £20.70 to standard form (2 s.f.):

   • 20.70 needs to be A x 10^n where 1 <= A < 10. So, 2.07 x 10^1.
   • Rounding 2.07 to 2 significant figures: the second significant figure is 0, and the next digit is 7 (which is 5 or more), so round up the 0 to 1.
   • Final answer: 2.1 x 10^1

4. Key Takeaways

• Always find a common denominator for adding/subtracting fractions, and remember to "flip and multiply" for division.
• Percentages are just fractions out of 100; convert them to decimals (/100) for calculations.
• Standard form uses A x 10^n where A is between 1 and 10; positive n means a large number, negative n means a small number.
• Estimation by rounding to one significant figure is a quick way to check if your answer is sensible.
• Be comfortable converting between fractions, decimals, and percentages.
• When calculating with percentages, be clear whether you're finding a percentage of an amount, an increase/decrease, or a reverse percentage.

• Standard form exponents add when multiplying and subtract when dividing.

• Common Mistakes:

  • Not finding a common denominator for fraction addition/subtraction.
  • Forgetting to "flip" the second fraction when dividing.
  • Confusing percentage increase/decrease with reverse percentages.
  • Incorrectly placing the decimal point or assigning the wrong sign to n in standard form.
  • Rounding incorrectly for estimation or final answers, especially with significant figures.

5. Now Try It

Spend 15 minutes working through this problem: A factory produces 25,000 components per day. 3/8 of these are sent to Europe, and 45% of the remainder are