Number
Integers, fractions, decimals, percentages, powers, roots, and standard form.
In this topic
Integers and Decimals
Integers are whole numbers — positive, negative, or zero. They form the foundation of all number work in GCSE Maths. Understanding place value is essential: each digit in a number has a value determined by its position.
When ordering decimals, compare digit by digit from left to right. For example, 0.35 is greater than 0.309 because 5 hundredths > 0 hundredths when we compare the second decimal place.
Key Points
- Place value determines the value of each digit
- Negative numbers follow rules: negative × negative = positive
- BIDMAS/BODMAS gives the order of operations
- Rounding: look at the next digit — 5 or more rounds up
Example Questions
2Put these numbers in order from smallest to largest: 0.7, 0.07, 0.71, 0.077
0.07, 0.077, 0.7, 0.71 — Compare digit by digit from the left after the decimal point.
[2 marks]
2Calculate: -3 × (-5) + 2 × (-4)
-3 × (-5) = 15, then 2 × (-4) = -8, so 15 + (-8) = 7
[2 marks]
1Round 3.0649 to 2 decimal places.
3.06 — The third decimal digit is 4, which is less than 5, so we round down.
[1 mark]
1Calculate 7.5 + 3.2.
7.5 + 3.2 = 10.7
[1 mark]
1What is the result of 15 - 8.6?
15 - 8.6 = 6.4
[1 mark]
2Show that 0.75 is equivalent to 3/4.
0.75 = 75/100 = 3/4 after simplifying by dividing both the numerator and denominator by 25.
[2 marks]
3A store sells a shirt for £24. The shirt is on sale for 15% off. Calculate the sale price.
15% of £24 = 0.15 * 24 = £3.6. Sale price = £24 - £3.6 = £20.4.
[3 marks]
3If a rectangle has a length of 10.5 cm and a width of 4.2 cm, find the area.
Area = length * width = 10.5 * 4.2 = 44.1 cm².
[3 marks]
4Prove that the sum of the integers from 1 to 20 is 210.
Sum = n/2 * (first number + last number) = 20/2 * (1 + 20) = 10 * 21 = 210.
[4 marks]
5A car travels 150.75 miles using 5.5 gallons of fuel. Calculate the fuel efficiency in miles per gallon (mpg) and explain your method.
Fuel efficiency = distance/fuel used = 150.75 miles / 5.5 gallons = 27.3863 mpg (approximately 27.39 mpg). Method: divide the total miles by the total gallons.
[5 marks]
Place Value
Each position in a number represents a power of 10. Moving left, each position is worth 10 times more. Moving right past the decimal point, each position is worth 10 times less.
For example, in the number 4,572.36: - 4 is in the thousands place (4 × 1000) - 5 is in the hundreds place (5 × 100) - 7 is in the tens place (7 × 10) - 2 is in the ones place (2 × 1) - 3 is in the tenths place (3 × 0.1) - 6 is in the hundredths place (6 × 0.01)
Key Points
- Each place is 10× the place to its right
- The decimal point separates whole numbers from fractions
Example Questions
1What is the value of the digit 7 in 37,429?
7,000 (seven thousand) — the 7 is in the thousands column.
[1 mark]
1What is the place value of the digit 7 in the number 47,258?
The place value of the digit 7 is 7,000.
[1 mark]
1Round the number 6,742 to the nearest hundred.
6,742 rounded to the nearest hundred is 6,700.
[1 mark]
2Write the number 0.0065 in standard form.
0.0065 in standard form is 6.5 x 10^-3.
[2 marks]
3Find the difference between the place value of 3 in 3,482 and the place value of 3 in 23,456.
The place value of 3 in 3,482 is 3,000 and in 23,456 is 30. The difference is 3,000 - 30 = 2,970.
[3 marks]
4A number has a digit 4 in the tens place and a digit 5 in the hundreds place. If the number is 4,582, explain how the place values of the digits contribute to the total value of the number.
The place value of 4 in the thousands is 4,000, the place value of 5 in the hundreds is 500, and the value of 8 in the tens is 80. Therefore, the total value is 4,000 + 500 + 80 + 2 = 4,582.
[4 marks]
4Show that the number 54,321 can be expressed as the sum of its place values.
54,321 = 50,000 + 4,000 + 300 + 20 + 1.
[4 marks]
2If you have the decimal number 12.345, what is the value of the digit in the hundredths place?
The digit in the hundredths place is 4, so its value is 0.04.
[2 marks]
3A number is represented as 7.8901 x 10^3. Calculate the number and identify the place value of the digit 9.
The number is 7890.1. The place value of the digit 9 is 900.
[3 marks]
5The number 1,234,567 is increased by 10 times. Write the new number and explain how the place values of the digits have changed.
The new number is 12,345,670. Each digit has shifted one place to the left, so the place value of each digit has increased by ten times.
[5 marks]
Negative Numbers
Negative numbers are less than zero. They appear on the left side of a number line. Key rules:
Multiplication and Division: - Positive × Positive = Positive - Negative × Negative = Positive - Positive × Negative = Negative - Negative × Positive = Negative
Addition and Subtraction: - Adding a negative is the same as subtracting: 5 + (-3) = 5 - 3 = 2 - Subtracting a negative is the same as adding: 5 - (-3) = 5 + 3 = 8
Key Points
- Two negatives make a positive when multiplying or dividing
- Subtracting a negative = adding a positive
- On a number line, left = smaller, right = bigger
Example Questions
1The temperature at midnight was -6°C. By noon it had risen by 11°C. What was the temperature at noon?
-6 + 11 = 5°C
[1 mark]
2Calculate (-8) ÷ (-2) × (-3)
(-8) ÷ (-2) = 4, then 4 × (-3) = -12
[2 marks]
1Calculate -7 + 5.
-7 + 5 = -2
[1 mark]
1What is -12 - (-4)?
-12 - (-4) = -12 + 4 = -8
[1 mark]
2A temperature drops from -3 degrees to -10 degrees. How many degrees has it dropped?
Temperature drop = -10 - (-3) = -10 + 3 = -7 degrees.
[2 marks]
2Show that -15 + 8 - 4 = -11.
-15 + 8 - 4 = -15 + 8 = -7; -7 - 4 = -11.
[2 marks]
3If x = -6, calculate the value of 3x + 2.
3(-6) + 2 = -18 + 2 = -16.
[3 marks]
3A bank account has a balance of -20. If a payment of 15 is made to this account, what will be the new balance?
New balance = -20 + 15 = -5.
[3 marks]
4Prove that the product of two negative numbers is positive by calculating (-4) × (-3).
(-4) × (-3) = 12, which is positive. Therefore, the product of two negative numbers is positive.
[4 marks]
5Find the value of x if -2(x + 5) = -16.
First, divide both sides by -2: x + 5 = 8. Then, subtract 5 from both sides: x = 3.
[5 marks]
Order of Operations (BIDMAS)
BIDMAS tells you the order to perform calculations: - B: Brackets first - I: Indices (powers and roots) - D: Division } left to right - M: Multiplication } left to right - A: Addition } left to right - S: Subtraction } left to right
Division and Multiplication are equal priority — work left to right. Addition and Subtraction are equal priority — work left to right.
Key Points
- Always start with brackets
- Indices come before ×, ÷, +, -
- × and ÷ are done before + and -
- Equal priority operations: work left to right
Example Questions
2Calculate: 3 + 4 × 2 - (6 ÷ 3)
Brackets first: 6 ÷ 3 = 2. Then multiplication: 4 × 2 = 8. Then left to right: 3 + 8 - 2 = 9
[2 marks]
1Calculate 8 + 2 × 5
8 + (2 × 5) = 8 + 10 = 18
[1 mark]
2Evaluate 6 × (4 + 2) - 10
6 × (4 + 2) - 10 = 6 × 6 - 10 = 36 - 10 = 26
[2 marks]
2Simplify 3 + 5 × 2 - 4 ÷ 2
3 + (5 × 2) - (4 ÷ 2) = 3 + 10 - 2 = 11
[2 marks]
3Show that 12 - 3 × (2 + 1) = 3
12 - 3 × (2 + 1) = 12 - 3 × 3 = 12 - 9 = 3, therefore it is shown.
[3 marks]
3A = 2 + 3 × 4 - 5 ÷ 5. Calculate the value of A.
A = 2 + (3 × 4) - (5 ÷ 5) = 2 + 12 - 1 = 13.
[3 marks]
4If B = 15 ÷ (3 + 2) × 4, find the value of B.
B = 15 ÷ (3 + 2) × 4 = 15 ÷ 5 × 4 = 3 × 4 = 12.
[4 marks]
4Prove that 5 × (2 + 6) ÷ 2 - 10 = 6.
5 × (2 + 6) ÷ 2 - 10 = 5 × 8 ÷ 2 - 10 = 40 ÷ 2 - 10 = 20 - 10 = 10, therefore the statement is false. Correct answer is 10.
[4 marks]
5Calculate the value of (2 + 3)² - (6 ÷ 2) × 3 + 5.
(2 + 3)² - (6 ÷ 2) × 3 + 5 = 5² - (3)×3 + 5 = 25 - 9 + 5 = 21.
[5 marks]
6Find the value of (3 × (2 + 5) - 4²) ÷ (2 + 1) + 6.
(3 × (2 + 5) - 4²) ÷ (2 + 1) + 6 = (3 × 7 - 16) ÷ 3 + 6 = (21 - 16) ÷ 3 + 6 = 5 ÷ 3 + 6 = 1.67 + 6 = 7.67.
[6 marks]
Fractions
A fraction represents a part of a whole. The top number (numerator) tells you how many parts you have. The bottom number (denominator) tells you how many equal parts the whole is divided into.
Equivalent fractions have the same value: 1/2 = 2/4 = 3/6. To simplify a fraction, divide both numerator and denominator by their highest common factor (HCF).
Key Points
- To add/subtract fractions, find a common denominator
- To multiply fractions, multiply numerators and denominators
- To divide by a fraction, flip it and multiply (KFC: Keep, Flip, Change)
- Mixed numbers can be converted to improper fractions
Example Questions
2Calculate 2/3 + 3/4
Common denominator = 12. So 8/12 + 9/12 = 17/12 = 1 5/12
[2 marks]
2Calculate 3/5 ÷ 2/7
Keep 3/5, flip 2/7 to 7/2, multiply: 3/5 × 7/2 = 21/10 = 2 1/10
[2 marks]
1Simplify 36/48 to its lowest terms.
HCF of 36 and 48 is 12. 36÷12 = 3, 48÷12 = 4. Answer: 3/4
[1 mark]
1What is 3/4 + 1/2?
To add the fractions, convert 1/2 to a fraction with a denominator of 4: 1/2 = 2/4. Now, 3/4 + 2/4 = 5/4 or 1 1/4.
[1 mark]
1Simplify the fraction 12/16.
Divide both the numerator and the denominator by their greatest common divisor (4): 12 ÷ 4 = 3 and 16 ÷ 4 = 4. The simplified fraction is 3/4.
[1 mark]
2A recipe requires 3/5 of a cup of sugar. If you want to make half of the recipe, how much sugar do you need?
To find half of 3/5, multiply by 1/2: (3/5) * (1/2) = 3/10. You need 3/10 of a cup of sugar.
[2 marks]
2Show that 2/3 - 1/6 = 3/6.
First, convert 2/3 to have a common denominator of 6: 2/3 = 4/6. Now subtract: 4/6 - 1/6 = 3/6, which simplifies to 1/2.
[2 marks]
3Find the value of x if 2/5x = 6.
To find x, first multiply both sides by 5 to eliminate the fraction: 2x = 6 * 5, which gives 2x = 30. Now divide both sides by 2: x = 30/2 = 15.
[3 marks]
4A class has 30 students. 1/3 of them are girls. If 4 girls leave and then 2 boys join, what fraction of the class are girls now?
First, find the number of girls: 1/3 of 30 is 10. After 4 girls leave, there are 10 - 4 = 6 girls. The total number of students is now 30 - 4 + 2 = 28. The fraction of girls is 6/28, which simplifies to 3/14.
[4 marks]
5A car can travel 3/5 of a mile on 1/4 of a gallon of fuel. How many miles can it travel on 1 gallon of fuel? Show your working.
If 1/4 gallon allows the car to travel 3/5 miles, then for 1 gallon (which is 4 times 1/4 gallon), the car can travel 4 * (3/5) = 12/5 miles, which simplifies to 2 2/5 miles.
[5 marks]
Adding and Subtracting Fractions
To add or subtract fractions, they must have the same denominator. Find the lowest common multiple (LCM) of the denominators.
Step 1: Find the LCM of the denominators Step 2: Convert each fraction to an equivalent fraction with the LCM as denominator Step 3: Add or subtract the numerators Step 4: Simplify if possible
Key Points
- Same denominator = just add/subtract numerators
- Different denominators = find LCM first
- Always simplify your answer
Example Questions
2Calculate 5/6 - 1/4
LCM of 6 and 4 is 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 - 3/12 = 7/12
[2 marks]
1Calculate 1/3 + 1/6.
To add the fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. So, convert 1/3 to 2/6. Now, 2/6 + 1/6 = 3/6 = 1/2.
[1 mark]
1Simplify the fraction 4/8 - 1/4.
First, convert 1/4 to an equivalent fraction with a denominator of 8. This gives us 2/8. Now, 4/8 - 2/8 = 2/8, which simplifies to 1/4.
[1 mark]
2Find the sum of 2/5 + 3/10.
The common denominator is 10. Convert 2/5 to 4/10. Now, 4/10 + 3/10 = 7/10.
[2 marks]
2Show that 5/6 - 1/3 = 1/2.
Convert 1/3 to 2/6. Now, 5/6 - 2/6 = 3/6 = 1/2.
[2 marks]
3Calculate 3/4 + 1/2 - 1/8.
First, convert 1/2 to 4/8. Now, find a common denominator for 3/4 (6/8) and add: 6/8 + 4/8 - 1/8 = 9/8 = 1 1/8.
[3 marks]
3A pizza was cut into 12 equal slices. If you eat 3/4 of the pizza, how many slices are left? Show your working.
3/4 of 12 slices is 9 slices. Therefore, 12 - 9 = 3 slices are left.
[3 marks]
4Explain why 2/3 - 1/4 cannot be simplified directly without finding a common denominator.
To subtract fractions, they must have the same denominator. Since 3 and 4 are not the same, we need to find a common denominator (which is 12) to perform the operation correctly.
[4 marks]
4Find the total of 1/2 + 1/3 + 1/6.
The common denominator is 6. Convert 1/2 to 3/6 and 1/3 to 2/6. Now, 3/6 + 2/6 + 1/6 = 6/6 = 1.
[4 marks]
5A recipe requires 2/3 cup of sugar and 1/4 cup of brown sugar. How much sugar is needed in total? Show your working clearly.
To add, find a common denominator. The common denominator of 3 and 4 is 12. Convert: 2/3 = 8/12 and 1/4 = 3/12. Now, 8/12 + 3/12 = 11/12 cups of sugar.
[5 marks]
Multiplying and Dividing Fractions
Multiplying fractions: Multiply numerators together, multiply denominators together, then simplify.
Dividing fractions (KFC method): - Keep the first fraction the same - Flip the second fraction (find its reciprocal) - Change the ÷ sign to ×
With mixed numbers, always convert to improper fractions first.
Key Points
- Multiply straight across: top × top, bottom × bottom
- You can cross-cancel before multiplying to keep numbers smaller
- Division = multiply by the reciprocal
Example Questions
3Calculate 2 1/3 × 1 2/5
Convert: 7/3 × 7/5 = 49/15 = 3 4/15
[3 marks]
1Calculate 2/3 × 4/5.
2/3 × 4/5 = (2 × 4) / (3 × 5) = 8/15.
[1 mark]
1Calculate 1/4 ÷ 2/3.
1/4 ÷ 2/3 = 1/4 × 3/2 = (1 × 3) / (4 × 2) = 3/8.
[1 mark]
2Show that 3/5 × 1/2 = 3/10.
3/5 × 1/2 = (3 × 1) / (5 × 2) = 3/10.
[2 marks]
2Find the value of 5/6 ÷ 1/3.
5/6 ÷ 1/3 = 5/6 × 3/1 = (5 × 3) / (6 × 1) = 15/6 = 5/2 or 2 1/2.
[2 marks]
3A recipe requires 2/3 cup of sugar. If you want to make 3 batches of the recipe, how much sugar do you need in total?
2/3 × 3 = 6/3 = 2 cups of sugar.
[3 marks]
4A bottle is filled with 3/4 liters of juice. If you pour out 1/2 of the juice, how much juice is left in the bottle?
1/2 × 3/4 = 3/8 liters poured out. Juice left = 3/4 - 3/8 = 6/8 - 3/8 = 3/8 liters.
[4 marks]
3Explain why multiplying 3/7 by 4/5 gives a fraction smaller than both 3/7 and 4/5.
When multiplying two proper fractions (both less than 1), the product is always smaller than either fraction. 3/7 × 4/5 = 12/35, which is less than both 3/7 and 4/5.
[3 marks]
4A gardener has 5/8 of a bag of soil. He uses 1/4 of the remaining soil. How much soil does he have left after using it?
Soil used = 1/4 × (5/8) = 5/32. Remaining soil = 5/8 - 5/32 = 20/32 - 5/32 = 15/32 bags of soil left.
[4 marks]
5A recipe calls for 3/4 of a cup of milk. You want to double the recipe and then divide it equally between 3 people. How much milk does each person get?
Double the recipe: 3/4 × 2 = 3/2 cups. Divide between 3 people: (3/2) ÷ 3 = (3/2) × (1/3) = 3/6 = 1/2 cup per person.
[5 marks]
Percentages
Percentage means 'out of 100'. To convert between fractions, decimals, and percentages: - Fraction → Decimal: divide numerator by denominator - Decimal → Percentage: multiply by 100 - Percentage → Decimal: divide by 100
Finding a percentage of an amount: 15% of 240 → 0.15 × 240 = 36
Key Points
- Percentage increase: new = original × (1 + rate/100)
- Percentage decrease: new = original × (1 - rate/100)
- Reverse percentages: to find original, divide by the multiplier
- Compound interest uses repeated percentage increase
Example Questions
2A jacket costs £85. It is reduced by 20% in a sale. What is the sale price?
£85 × 0.80 = £68
[2 marks]
2After a 15% increase, a house is worth £230,000. What was it worth before the increase?
£230,000 ÷ 1.15 = £200,000
[2 marks]
3£5000 is invested at 3% compound interest per year. What is it worth after 4 years?
£5000 × 1.03⁴ = £5000 × 1.1255... = £5627.54
[3 marks]
1What is 25% of 200?
25% of 200 = 0.25 * 200 = 50
[1 mark]
2A jacket costs £60. It is on sale for 20% off. What is the sale price of the jacket?
20% of £60 = 0.2 * 60 = £12. Sale price = £60 - £12 = £48.
[2 marks]
3A population of a town increased from 10,000 to 12,500. Calculate the percentage increase.
Increase = 12,500 - 10,000 = 2,500. Percentage increase = (2,500 / 10,000) * 100 = 25%.
[3 marks]
2A student scored 45 out of 60 on a test. What percentage did the student score?
Percentage score = (45 / 60) * 100 = 75%.
[2 marks]
4A shop sells a pair of shoes for £80. If the shoes are subject to a 15% VAT increase, what will be the final price?
VAT amount = 15% of £80 = 0.15 * 80 = £12. Final price = £80 + £12 = £92.
[4 marks]
4A car's value depreciates by 10% each year. If the car is currently worth £20,000, what will its value be at the end of 2 years?
Value after 1 year = £20,000 - (10% of £20,000) = £20,000 - £2,000 = £18,000. Value after 2 years = £18,000 - (10% of £18,000) = £18,000 - £1,800 = £16,200.
[4 marks]
5Show that if a product costs £120 and is discounted by 30%, the final price is £84.
30% of £120 = 0.30 * 120 = £36. Final price = £120 - £36 = £84. Therefore, it is shown that the final price after the discount is £84.
[5 marks]
Percentage Change
Percentage change = (change ÷ original) × 100
This formula works for both increases and decreases. If the result is positive, it's an increase. If negative, it's a decrease.
Example: A phone was £400, now £340. Change = 400 - 340 = 60 Percentage change = (60 ÷ 400) × 100 = 15% decrease
Key Points
- Always divide by the ORIGINAL value
- State whether it's an increase or decrease
Example Questions
2A car was bought for £12,000 and sold for £9,600. Find the percentage loss.
Loss = £2,400. Percentage = (2400 ÷ 12000) × 100 = 20% loss
[2 marks]
2A product costs £80 and is reduced by 15%. What is the new price of the product?
The reduction is 15% of £80, which is 0.15 * 80 = £12. The new price is £80 - £12 = £68.
[2 marks]
2The population of a town is 20,000. If the population increases by 10%, what is the new population?
10% of 20,000 is 0.10 * 20,000 = 2,000. The new population is 20,000 + 2,000 = 22,000.
[2 marks]
2A bike was originally priced at £500. During a sale, the price increased by 20%. What is the new price of the bike?
20% of £500 is 0.20 * 500 = £100. The new price is £500 + £100 = £600.
[2 marks]
2A student scored 45 out of 60 on a test. What is the percentage score? Show your workings.
Percentage score = (45/60) * 100 = 75%.
[2 marks]
3The price of a jacket is increased from £60 to £72. Calculate the percentage increase in the price.
Increase = £72 - £60 = £12. Percentage increase = (12/60) * 100 = 20%.
[3 marks]
3A car's value depreciates from £10,000 to £7,500. Calculate the percentage decrease in the value of the car.
Decrease = £10,000 - £7,500 = £2,500. Percentage decrease = (2500/10000) * 100 = 25%.
[3 marks]
4A shop increased the price of a TV by 30% to £520. What was the original price of the TV? Show all workings.
Let the original price be x. Then, x + 0.30x = 520. This simplifies to 1.30x = 520. Therefore, x = 520/1.30 = £400.
[4 marks]
4A company reported a profit of £200,000 last year. This year, the profit increased by 15%. Calculate the profit for this year.
Increase = 15% of £200,000 = 0.15 * 200,000 = £30,000. This year’s profit = £200,000 + £30,000 = £230,000.
[4 marks]
5A store sells a smartphone for £600 after a 25% discount. Prove that the original price of the smartphone was £800.
Let the original price be x. The discount is 25% of x, so the price after discount is x - 0.25x = 0.75x. Setting up the equation, 0.75x = 600. Thus, x = 600/0.75 = £800.
[5 marks]
Compound Interest and Depreciation
Compound interest means interest is added to the principal, and future interest is earned on the new total.
Formula: Final amount = P × (1 + r/100)ⁿ Where P = principal, r = rate per period, n = number of periods
For depreciation (losing value): Final amount = P × (1 - r/100)ⁿ
Key Points
- Compound = interest on interest
- Depreciation uses (1 - r/100) instead of (1 + r/100)
- n is the number of time periods, not years necessarily
Example Questions
3A car worth £18,000 depreciates by 12% each year. What is it worth after 3 years?
£18,000 × (1 - 0.12)³ = £18,000 × 0.88³ = £18,000 × 0.681472 = £12,266.50
[3 marks]
2Calculate the amount of money you will have after 3 years if you invest £1000 at an interest rate of 5% per annum compounded annually.
A = P(1 + r)^n = 1000(1 + 0.05)^3 = 1000(1.157625) = £1157.63
[2 marks]
2A car is bought for £15,000. It depreciates in value by 10% each year. What will be the value of the car after 2 years?
Value after 1 year: 15000 * (1 - 0.10) = 13500. Value after 2 years: 13500 * (1 - 0.10) = 12150. The value of the car after 2 years is £12,150.
[2 marks]
3An investment of £5000 earns 4% interest compounded annually. How much will the investment be worth after 5 years?
A = 5000(1 + 0.04)^5 = 5000(1.2166529) = £6083.26.
[3 marks]
3A laptop is purchased for £800. It depreciates at a rate of 20% per year. Show that its value after 3 years is approximately £409.60.
Value after 1 year: 800 * (1 - 0.20) = 640. Value after 2 years: 640 * (1 - 0.20) = 512. Value after 3 years: 512 * (1 - 0.20) = 409.60.
[3 marks]
3You invest £1200 in a savings account that offers an interest rate of 6% compounded annually. How much money will you have after 4 years?
A = 1200(1 + 0.06)^4 = 1200(1.262477) = £1514.97.
[3 marks]
4A house is worth £250,000 and depreciates by 5% each year. Calculate the value of the house after 4 years.
Value after 1 year: 250000 * (1 - 0.05) = 237500. After 2 years: 237500 * (1 - 0.05) = 225625. After 3 years: 225625 * (1 - 0.05) = 214343.75. After 4 years: 214343.75 * (1 - 0.05) = £203626.56.
[4 marks]
4An investment of £2000 earns interest at a rate of 8% per annum, compounded annually. How much interest will be earned in total after 2 years?
Total amount after 2 years: A = 2000(1 + 0.08)^2 = 2000(1.1664) = 2332.80. Interest earned = 2332.80 - 2000 = £332.80.
[4 marks]
5A company purchased a machine for £50,000. The machine depreciates at a rate of 15% per year. Calculate the value of the machine after 5 years. Then, find how much total depreciation has occurred.
Value after 5 years: A = 50000(1 - 0.15)^5 = 50000(0.4374) = £21870. Total depreciation = 50000 - 21870 = £28130.
[5 marks]
6A bank offers a savings account with a compound interest rate of 7% per annum. You deposit £10,000 and leave it for 6 years. Calculate the total amount in the account at the end of the 6 years. Show all your workings.
A = 10000(1 + 0.07)^6. A = 10000(1.484812) = £14848.12. The total amount in the account after 6 years is £14,848.12.
[6 marks]
Powers and Roots
A power (index) tells you how many times to multiply a number by itself. For example, 2³ = 2 × 2 × 2 = 8.
Square root (√) is the inverse of squaring. Cube root (∛) is the inverse of cubing.
Laws of indices: - aᵐ × aⁿ = aᵐ⁺ⁿ - aᵐ ÷ aⁿ = aᵐ⁻ⁿ - (aᵐ)ⁿ = aᵐˣⁿ - a⁰ = 1 - a⁻ⁿ = 1/aⁿ - a^(1/n) = ⁿ√a
Key Points
- Anything to the power of 0 equals 1
- Negative indices mean reciprocals
- Fractional indices mean roots
- When multiplying same bases, add the powers
Example Questions
1Simplify 2³ × 2⁵
2³⁺⁵ = 2⁸ = 256
[1 mark]
2Evaluate 27^(2/3)
27^(1/3) = 3 (cube root), then 3² = 9
[2 marks]
2Simplify (3x²y)³
3³ × x²ˣ³ × y³ = 27x⁶y³
[2 marks]
1Calculate 5².
5² = 25.
[1 mark]
1Find the value of √144.
√144 = 12.
[1 mark]
2Simplify the expression (2³ × 2²).
2³ × 2² = 2^(3+2) = 2⁵ = 32.
[2 marks]
2Show that 3² + 4² = 25.
3² = 9, 4² = 16, so 3² + 4² = 9 + 16 = 25.
[2 marks]
3If x = 3, evaluate 2x² - 5.
2(3)² - 5 = 2(9) - 5 = 18 - 5 = 13.
[3 marks]
3Find the value of x if x² = 64.
x = √64 = 8 or x = -√64 = -8. So, x = 8 or -8.
[3 marks]
5Prove that (2² × 3²) × (3 × 2) = 6².
(2² × 3²) × (3 × 2) = (4 × 9) × (6) = 36 × 6 = 216. 6² = 36, therefore equality doesn't hold; thus, the statement is false.
[5 marks]
Standard Form
Standard form is a way to write very large or very small numbers concisely. A number in standard form looks like: a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
Large numbers have a positive power of 10: - 5,600,000 = 5.6 × 10⁶
Small numbers have a negative power of 10: - 0.00034 = 3.4 × 10⁻⁴
Key Points
- a must be between 1 and 10 (including 1, not including 10)
- Count decimal places moved = the power
- Moving left = positive power; moving right = negative power
- To add/subtract, convert to same power of 10 first
Example Questions
1Write 0.00052 in standard form.
5.2 × 10⁻⁴
[1 mark]
2Calculate (3 × 10⁴) × (2 × 10⁵). Give your answer in standard form.
3 × 2 = 6, 10⁴ × 10⁵ = 10⁹. Answer: 6 × 10⁹
[2 marks]
1Write the number 45000 in standard form.
4.5 x 10^4
[1 mark]
1Convert the number 0.00056 into standard form.
5.6 x 10^-4
[1 mark]
2Calculate the product of 3.2 x 10^3 and 4 x 10^2 in standard form.
1.28 x 10^6
[2 marks]
3A bacteria culture doubles in size every hour. If the initial population is 1.5 x 10^3, what will be the population after 5 hours, in standard form?
4.8 x 10^4
[3 marks]
3Show that (2 x 10^3) + (3.5 x 10^2) = 2.35 x 10^3, giving your answer in standard form.
2 x 10^3 + 0.35 x 10^3 = 2.35 x 10^3
[3 marks]
4A car travels 3.6 x 10^5 meters in a day. How far does it travel in 30 days? Give your answer in standard form.
1.08 x 10^7 meters
[4 marks]
4Find the value of (5 x 10^4) / (2 x 10^2) and express your answer in standard form.
2.5 x 10^2
[4 marks]
5Prove that the area of a circle with a radius of 3 x 10^2 cm is approximately 2.83 x 10^5 cm², using standard form. (Use π ≈ 3.14)
Area = πr² = 3.14 x (3 x 10^2)² = 3.14 x 9 x 10^4 = 28.26 x 10^4 = 2.83 x 10^5 cm²
[5 marks]
Ratio and Proportion
A ratio compares parts to parts. A proportion compares a part to the whole.
Simplifying ratios: divide all parts by their HCF. For example, 12:8 → 3:2 (divide by 4).
Sharing in a ratio: Add the parts, divide the total, then multiply. To share £100 in the ratio 3:2: total parts = 5, one part = £20, so 3 parts = £60 and 2 parts = £40.
Key Points
- Ratios must be in the same units before simplifying
- Direct proportion: as one increases, the other increases at the same rate
- Inverse proportion: as one increases, the other decreases
- Unitary method: find the value of 1, then scale
Example Questions
2Share £450 in the ratio 2:3:4
Total parts = 9. One part = £50. Shares: £100, £150, £200
[2 marks]
2If 5 pens cost £3.75, how much do 8 pens cost?
1 pen = £3.75 ÷ 5 = £0.75. 8 pens = 8 × £0.75 = £6.00
[2 marks]
1A recipe requires 3 cups of flour for every 2 cups of sugar. What is the ratio of flour to sugar?
The ratio of flour to sugar is 3:2.
[1 mark]
2If a car travels 150 miles using 5 gallons of fuel, how many miles can it travel using 8 gallons of fuel? Show your working.
The car travels 150 miles / 5 gallons = 30 miles per gallon. Therefore, for 8 gallons, it travels 30 miles/gallon * 8 gallons = 240 miles.
[2 marks]
3In a class, the ratio of boys to girls is 4:5. If there are 36 girls, how many boys are there?
Let the number of boys be 4x and girls be 5x. 5x = 36, so x = 36/5 = 7.2. Therefore, boys = 4 * 7.2 = 28.8. Since we can't have a fraction of a boy, the answer must be checked or the question rephrased to integers. Assume a mistake in the ratio for integer values: Boys = 28.
[3 marks]
3A scale model of a building is made with a ratio of 1:50. If the height of the model is 2 meters, what is the actual height of the building? Show your calculations.
If the model is 1/50 of the actual height, the actual height = model height * scale factor = 2 meters * 50 = 100 meters.
[3 marks]
4A bag contains red and blue marbles in the ratio 3:2. If there are 25 marbles in total, how many red marbles are there? Show all workings.
Let the number of red marbles be 3x and blue marbles be 2x. Thus, 3x + 2x = 25, giving 5x = 25. Therefore, x = 5. Red marbles = 3x = 3 * 5 = 15.
[4 marks]
4Prove that if the ratio of two quantities A and B is 4:3, and the sum of A and B is 70, then A is 40 and B is 30.
Let A = 4x and B = 3x. Thus, 4x + 3x = 70, giving 7x = 70. Therefore, x = 10. Thus, A = 4 * 10 = 40 and B = 3 * 10 = 30, which proves the statement.
[4 marks]
5A recipe requires the ingredients in the ratio of 5:3:2. If you need a total of 100 grams of the mixture, how much of each ingredient do you need? Show your workings.
The total parts are 5 + 3 + 2 = 10 parts. Each part is 100 grams / 10 parts = 10 grams. Therefore, ingredient 1 = 5 * 10 = 50 grams, ingredient 2 = 3 * 10 = 30 grams, ingredient 3 = 2 * 10 = 20 grams.
[5 marks]
6A fruit seller sells apples and oranges in the ratio of 7:5. During a sale, he sold 200 fruits in total. If he sold 80 apples, how many oranges did he sell? Show your reasoning.
Let the number of apples be 7x and oranges be 5x. We know total fruits sold = 7x + 5x = 12x = 200. Thus, x = 200/12 = 16.67. Apples sold = 80, which is 7x = 7 * 16.67 = 116.67 (rounding error indicates misstatement). Assuming consistent ratio: If 80 apples = 7 parts, oranges = 5 parts = (5/7) * 80 = 57.14. Must adjust to closest integer based on total. Thus, oranges = 120 - 80 = 120.
[6 marks]