The Four Operations
This topic covers the fundamental arithmetic skills required for all quantitative problems in the ESAT. Mastery of non-calculator operations on integers, decimals, and fractions is essential for both speed and accuracy.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- For decimal addition and subtraction, always align the numbers by their decimal points, not the last digit.
- When multiplying decimals, multiply the numbers as if they were integers, then count the total number of decimal places in the original numbers to position the decimal point in the answer.
- To divide by a decimal, multiply both the dividend (the number being divided) and the divisor by a power of 10 to make the divisor an integer.
- For fraction addition and subtraction, you must first find a common denominator.
- Always convert mixed numbers into improper (top-heavy) fractions before performing multiplication or division.
- Dividing by a fraction is equivalent to multiplying by its reciprocal (the fraction flipped upside down).
Formulae
(a/b) / (c/d) = a/b × d/c Use this rule to perform division involving fractions. Invert the second fraction (the divisor) and change the operation to multiplication.
Definitions
- Place Value
- The value of a digit based on its position within a number. For example, in 12.3, the '1' represents 10, the '2' represents 2 units, and the '3' represents 3 tenths.
- Improper Fraction
- A fraction where the numerator (top number) is greater than or equal to the denominator (bottom number), such as 7/3.
- Reciprocal
- The result of inverting a number or fraction. The reciprocal of x is 1/x, and the reciprocal of a/b is b/a.
Worked example
Calculate (0.4 × 15/2) - 1 1/4, giving your answer as a decimal.
- 1
Step 1:
Convert all numbers to a single format, either decimals or fractions.
Fractions are often easier for non-calculator work.
Convert 0.4 to 4/10 which simplifies to 2/5.
Convert the mixed number 1 1/4 to the improper fraction (1*4 + 1)/4 = 5/4.
- 2
Step 2:
Substitute these back into the expression:
(2/5 × 15/2) - 5/4.
- 3
Step 3:
Perform the multiplication first.
You can cancel common factors before multiplying:
(2 × 15) / (5 × 2) = 15/5 = 3.
- 4
Step 4:
The expression is now simplified to 3 - 5/4.
- 5
Step 5:
To subtract the fraction, find a common denominator.
3 can be written as 12/4.
The calculation becomes 12/4 - 5/4.
- 6
Step 6:
Perform the subtraction:
(12 - 5) / 4 = 7/4.
- 7
Step 7:
Convert the final answer to a decimal as requested.
7/4 = 1 3/4 = 1.75
Answer: 1.75
Common mistakes
- ×Mistakes with negative signs, especially in subtractions like 3 - (-5) or multiplications like (-4) × (-2).
- ×Misplacing the decimal point in multiplication. For example, writing 0.2 × 0.4 = 0.8 instead of the correct 0.08.
- ×Forgetting to find a common denominator for adding/subtracting fractions, and instead just adding the numerators and denominators.
- ×Incorrectly handling mixed numbers in multiplication/division, for instance by multiplying the integer parts and fractional parts separately.
No-calculator tips
- ✓Always estimate your answer before calculating. For 29.1 × 1.9, think 'approximately 30 × 2 = 60'. This helps you spot major errors, especially with decimal placement.
- ✓When multiplying and dividing fractions, cancel common factors from the numerators and denominators *before* you multiply. This keeps the numbers small and manageable.
- ✓Memorise common fraction-to-decimal conversions to save time, such as 1/4=0.25, 1/5=0.2, 1/8=0.125, and their multiples.
- ✓To multiply by 5, you can multiply by 10 and then halve the result. To divide by 5, you can divide by 10 and then double the result.