Fundamentals of Number: Operations, Fractions, and Percentages

Worked Example: Applying the Order of Operations

Worked Example: Following the BODMAS Code

1. 1
   First up, Brackets. We solve what's inside the $(...)$ first. Inside the brackets, we have an Order ($2^2$), so we do that.

   $2^2 = 4$
2. 2
   Now we finish the calculation inside the brackets: $3 \times 4$.

   $10 + (12) \div 6 - 4$
3. 3
   Brackets are done. Next is Division and Multiplication. We have a division: $12 \div 6$.

   $10 + 2 - 4$
4. 4
   Last up is Addition and Subtraction. We work from left to right. First, $10 + 2$.

   $12 - 4$
5. 5
   And the final step...

   $12 - 4 = 8$

Worked Example: Unifying Number Formats in Calculations

Worked Example: Unifying the Number Formats

1. 1
   Let's convert the mixed number to a decimal. We know that $\frac{1}{2}$ is the same as $0.5$.

   $3\frac{1}{2} = 3 + 0.5 = 3.5$
2. 2
   Now that both numbers are decimals, it's a simple addition problem.

   $3.5 + 1.75 = 5.25$
3. 3
   So, you used $5.25$ litres of milk. You could also write this as $5\frac{1}{4}$ litres.

   $Total = 5.25 \text{ litres}$

Worked Example: Ranking Different Number Formats

Worked Example: Ranking the Numbers

1. 1
   Let's convert everyone's battery level to a decimal so we can compare them easily.

   $\text{You: } \frac{4}{5} = 4 \div 5 = 0.80$
2. 2
   Next, convert Sarah's percentage to a decimal by dividing by 100.

   $\text{Sarah: } 85\% = 85 \div 100 = 0.85$
3. 3
   Ben's is already a decimal, so we don't need to change it. Now we compare the three decimals: $0.80$, $0.85$, and $0.82$.

   $0.85 > 0.82 > 0.80$
4. 4
   So, Sarah has the most battery. The order from largest to smallest is Sarah ($85\%$), then Ben ($0.82$), then you ($\frac{4}{5}$).

   $85\% > 0.82 > \frac{4}{5}$

Worked Example: Calculating a Percentage Discount

Worked Example: The Sneaker Discount

1. 1
   We need to calculate $30\%$ of $£120$. First, convert the percentage to a decimal. This is our multiplier.

   $30\% = 30 \div 100 = 0.30$
2. 2
   Now, multiply this decimal by the total cost of the sneakers.

   $0.30 \times £120 = £36$
3. 3
   Boom! You save $£36$. The final price would be $£120 - £36 = £84$. Score!

   $\text{Discount} = £36$

Worked Example: Calculating a Price Increase

Worked Example: Streaming Subscription Price Change

1. 1
   This is a percentage increase. We start with the original $100\%$ and add the $10\%$ increase.

   $100\% + 10\% = 110\%$
2. 2
   Convert this new percentage into a decimal multiplier.

   $110\% = 110 \div 100 = 1.10$
3. 3
   Multiply the original price by this multiplier to find the new price. Remember to round your answer to two decimal places for money.

   $£9.99 \times 1.10 = £10.989$
4. 4
   Rounding to the nearest penny, the new price is $£10.99$.

   $\text{New Price} = £10.99$

Worked Example: Calculating Percentage Growth

Worked Example: Follower Growth

1. 1
   First, let's find the 'Change' in followers. This is the new amount minus the original amount.

   $\text{Change} = 540 - 450 = 90$
2. 2
   Next, we use the percentage change formula. We divide the change (90) by the original number of followers (450).

   $\frac{90}{450} = 0.2$
3. 3
   Finally, multiply by 100 to turn the decimal into a percentage.

   $0.2 \times 100 = 20\%$
4. 4
   You had a $20\%$ increase in followers! Time to go viral. ✨

   $\text{Percentage Increase} = 20\%$

Worked Example: Finding the Original Value

Worked Example: Finding the Original Price

1. 1
   The final price includes the original $100\%$ plus a $10\%$ fee. So, the $£45$ represents $110\%$ of the original price.

   $100\% + 10\% = 110\%$
2. 2
   We can set up an equation. Let 'P' be the original price. We know that $110\%$ of P is $£45$. Let's write $110\%$ as a decimal.

   $1.10 \times P = £45$
3. 3
   To find P, we need to do the reverse of multiplying. We divide both sides by 1.10.

   $P = \frac{£45}{1.10} = £40.90909...$
4. 4
   Rounding to two decimal places for money, the original ticket price was $£40.91$.

   $\text{Original Price} = £40.91$

Worked Example: Applying Successive Discounts

Worked Example: The Double Discount

1. 1
   Let's find the decimal multipliers for both discounts. A $20\%$ discount means you pay $80\%$ ($100\% - 20\%$). A $10\%$ discount means you pay $90\%$ ($100\% - 10\%$).

   $\text{Multiplier 1} = 0.80 \quad \text{and} \quad \text{Multiplier 2} = 0.90$
2. 2
   Now, apply the first discount to the original price.

   $£500 \times 0.80 = £400$
3. 3
   This $£400$ is the sale price. Now apply the second discount to this new price.

   $£400 \times 0.90 = £360$
4. 4
   The final price is $£360$. Alternatively, you could multiply the multipliers together first ($0.8 \times 0.9 = 0.72$) and then multiply by the original price ($£500 \times 0.72 = £360$). Same result, fewer steps!

   $\text{Final Price} = £360$

Worked Example: Calculating Simple Interest Over Time

Worked Example: Savings from a Part-Time Job

1. 1
   Let's identify our P, R, and T from the problem.

   $P = £800, \quad R = 3\% = 0.03, \quad T = 4 \text{ years}$
2. 2
   Now, we plug these values into the simple interest formula: $I = P \times R \times T$.

   $I = 800 \times 0.03 \times 4$
3. 3
   Calculate the result.

   $I = £96$
4. 4
   After 4 years, you will have earned $£96$ in interest. The total amount in your account would be $£800 + £96 = £896$. Not bad for doing nothing!

   $\text{Interest Earned} = £96$

Worked Example: Prime Factorisation, HCF, and LCM

Worked Example: The Ultimate Party Plan 🥳

1. 1
   First, let's break down our numbers into their prime factors. This is like finding the basic DNA of 72 and 90. We'll use factor trees to make it visual.

   $72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2^3 \times 3^2$
   $90 = 2 \times 45 = 2 \times 3 \times 15 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5$
2. 2
   To find the greatest number of snack plates (part a), we need the Highest Common Factor (HCF). We look at the prime factors and multiply the lowest power of each factor that they both share.

   $\text{Common factors are } 2 \text{ and } 3^2.<br>\text{HCF} = 2^1 \times 3^2 = 2 \times 9 = 18$
   So, you can make a maximum of 18 identical snack plates!
3. 3
   Now for the prize draws (part b), we need to find when they'll sync up. This means finding the Lowest Common Multiple (LCM). To find the LCM, we multiply the highest power of all the prime factors from both numbers.

   $\text{Highest power of 2 is } 2^3.<br>\text{Highest power of 3 is } 3^2.<br>\text{Highest power of 5 is } 5^1.<br>\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$
4. 4
   Let's finalise the answers. The HCF gives us the answer for the snack plates, and the LCM gives us the answer for the prize draws.

   a) The greatest number of snack plates is 18. Each plate will have $72 \div 18 = 4$ sausage rolls and $90 \div 18 = 5$ spring rolls.
   b) The prize draws will happen at the same time after 360 seconds (or 6 minutes).

Worked Example: Calculating with Powers and Roots

The Ultimate Calculation Mashup ✨

1. 1
   Okay, let's break this expression down. It looks complicated, but it's just three mini-problems in one. We need to handle the square ($12^2$), the cube root ($\sqrt[3]{125}$), and the square root ($\sqrt{49}$) separately before we combine them.

   $(12^2) + (\sqrt[3]{125}) - (\sqrt{49})$
2. 2
   First up, the square. $12^2$ means $12 \times 12$. This is one of the key squares you need to memorise for the exam!

   $12^2 = 12 \times 12 = 144$
3. 3
   Next, the cube root. We're looking for the number that, when multiplied by itself three times, gives 125. Since it ends in a 5, the answer is probably 5. Let's check: $5 \times 5 \times 5 = 25 \times 5 = 125$. Perfect. This is another one to lock in your memory.

   $\sqrt[3]{125} = 5$
4. 4
   Last of the mini-problems: the square root. What number times itself equals 49? That's a classic from your times tables.

   $\sqrt{49} = 7$
5. 5
   Now we're on the final level. We just substitute our answers back into the original expression and solve it like a normal calculation. Follow the order of operations.

   $144 + 5 - 7 = 149 - 7 = 142$

Worked Example: Multi-Step Percentage Decrease

That Sale Price Isn't Gonna Calculate Itself 🛍️

1. 1
   First, let's figure out the initial 15% discount. The easiest way is to convert the percentage to a decimal and multiply it by the original price.

   $15\% = 0.15 \\ \text{Discount} = 0.15 \times \$80 = \$12$
2. 2
   Now, subtract that discount from the original price to find the first sale price. This is the price before your extra loyalty discount.

   $\text{Sale Price} = \$80 - \$12 = \$68$
3. 3
   Super important: The next 10% discount is taken from the new sale price ($68), not the original$80. Let's calculate this second discount.

   $10\% = 0.10 \\ \text{Second Discount} = 0.10 \times \$68 = \$6.80$
4. 4
   Finally, subtract the second discount from the sale price to get the final price you'll pay at the checkout. GG! 🎮

   $\text{Final Price} = \$68 - \$6.80 = \$61.20$

Worked Example: Comparing Financial Data

Sorting Out Your Friends' Account Balances 💸

1. 1
   First up, let's list all the financial amounts we need to compare. We have your balance, Sam's, Chloe's, and David's. One of them needs a quick calculation.

   Your Balance: $£15.50$
   Sam's Balance: $-£5$
   Chloe's Balance: $\frac{1}{2} \times £20$
   David's Balance: $-£5.25$
2. 2
   To make them easy to compare, let's make sure every value is in the same format. The only one that isn't a simple decimal or integer is Chloe's, so let's calculate that.

   $\text{Chloe's Balance} = \frac{1}{2} \times 20 = £10$
3. 3
   Now we have a clean list of decimals and integers. This is much easier to work with!

   The numbers to order are: $15.50, -5, 10, -5.25$
4. 4
   Let's think about a number line. The negative numbers are the smallest. The more negative a number is, the smaller it is. So, $-5.25$ is smaller than $-5$. Then we have the positive numbers, where $10$ is smaller than $15.50$.

   Smallest to Largest: $-5.25$, then $-5$, then $10$, then $15.50$
5. 5
   Finally, let's write it out as one continuous inequality, just like the question asks. This shows the final, ranked order of everyone's money.

   $-5.25 < -5 < 10 < 15.50$

Worked Example: Application of Standard Form

Calculating Your Spotify Stream Time 🎧

1. 1
   First, let's get both our numbers ready for the calculation. The number of streams is already in standard form: $1.5 \times 10^9$. We need to convert the song length, 200 seconds, into standard form. To do this, we move the decimal point two places to the left.

   $200 = 2.0 \times 10^2$
2. 2
   To find the total time, we need to multiply the number of streams by the length of one stream. Think of it like this: if you play a game for 30 minutes every day for 10 days, you'd multiply $30 \times 10$ to get the total time.

   $Total \; Time = (Number \; of \; Streams) \times (Length \; of \; Song)$ $Total \; Time = (1.5 \times 10^9) \times (2 \times 10^2)$
3. 3
   Now for the calculation. We multiply the front numbers together ($1.5 \times 2$) and then we add the powers of 10 ($9 + 2$). On Paper 3, you can just plug this straight into your calculator using the `x10^x` or `EXP` button, but it's good to know what's happening behind the scenes!

   $1.5 \times 2 = 3$ $10^9 \times 10^2 = 10^{9+2} = 10^{11}$
4. 4
   Let's put it all together. Our final answer is the product of the two parts. We check that the front number (3) is between 1 and 10, which it is, so our answer is in correct standard form. That's a lot of listening time!

   $Total \; Time = 3 \times 10^{11} \; seconds$

Calculating Bounds for a Measured Length

How Long Was That Drive, REALLY? 🚗

1. 1
   First, let's pull out the key info. We have the measured value and how accurately it was measured. This is our starting point.

   Measured Value = 25 km
   Accuracy = 1 km
2. 2
   This is the most important move! To find the 'wiggle room' on either side of 25 km, we take the accuracy (1 km) and chop it in half.

   $\frac{1}{2} = 0.5 \text{ km}$
3. 3
   To find the Lower Bound (the shortest the trip could possibly be), we subtract our 'wiggle room' value from the measured distance.

   $25 - 0.5 = 24.5 \text{ km}$
4. 4
   Now for the Upper Bound. We add the 'wiggle room' value to the measured distance. This is the value the actual distance gets right up to, but can't be equal to (or it would round up!).

   $25 + 0.5 = 25.5 \text{ km}$
5. 5
   Finally, we write this as an error interval using inequalities. We're saying the actual distance, $d$, is greater than or equal to the lower bound, but strictly less than the upper bound. Done! ✅

   $24.5 \le d < 25.5$

Worked Example: Ratio and Average Speed in Context

Solving a Road Trip Problem 🚗💨

1. 1
   First, let's tackle part (a), splitting the cost. We need to find the total number of parts in the ratio to figure out what one 'part' of the cost is worth.

   $Total\ Parts = 3 + 2 + 5 = 10\ parts$
2. 2
   Now, we divide the total cost by the total number of parts to find the value of a single part.

   $Value\ of\ 1\ part = \frac{Total\ Cost}{Total\ Parts} = \frac{£80}{10} = £8$
3. 3
   Finally for part (a), we multiply the value of one part by each person's number of parts in the ratio to find their individual shares.

   $Your\ share (3\ parts): 3 \times £8 = £24<br>Liam's\ share (2\ parts): 2 \times £8 = £16<br>Sara's\ share (5\ parts): 5 \times £8 = £40$
4. 4
   On to part (b)! We need to calculate the average speed. Let's start by writing down the formula.

   $Average\ Speed = \frac{Total\ Distance}{Total\ Time}$
5. 5
   Now we substitute the values given in the problem into our formula. The total distance is 360 km and the total time is 8 hours.

   $Average\ Speed = \frac{360\ km}{8\ hours}$
6. 6
   Last step! We just do the division to find the final answer. Don't forget the units!

   $Average\ Speed = 45\ km/h$

Worked Example: International Travel and Currency Conversion

The Ultimate Glow-Up: From Tokyo to London ✈️💸

1. 1
   First, let's figure out the arrival time in Tokyo's time zone. We do this by adding the flight duration to the departure time.

   $\text{Departure Time: } 11:30 \text{ (Tuesday)} \\ \text{Add flight duration: } + 14 \text{ hours } 15 \text{ minutes} \\ \text{Calculation: } 11:30 + 14h = 25:30. \text{ Then } 25:30 + 15m = 25:45 \\ \text{Since 24:00 is midnight, } 25:45 \text{ is } 01:45 \text{ the next day.} \\ \text{Arrival time (Tokyo time): } 01:45 \text{ on Wednesday}$
2. 2
   Now we need to adjust for the 9-hour time difference. London is 9 hours behind Tokyo, so we subtract 9 hours from the Tokyo arrival time.

   $\text{Tokyo arrival time: } 01:45 \text{ (Wednesday)} \\ \text{Subtract time difference: } - 9 \text{ hours} \\ \text{Let's count back: } 01:45 \text{ back 1h 45m is } 00:00 \text{ (start of Wed). We still need to go back } 7h \text{ 15m.} \\ 24:00 \text{ (end of Tue) } - 7 \text{ hours } 15 \text{ minutes} = 16:45 \text{ (Tuesday)} \\ \text{Arrival time (London time): } 16:45 \text{ on Tuesday}$
3. 3
   For the souvenir, we need to convert Japanese Yen (¥) to British Pounds (£). The rate is $£1 = ¥175$. To find how many pounds we get, we divide the amount in yen by the exchange rate.

   $\text{Cost in } £ = \frac{\text{Cost in } ¥}{\text{Exchange Rate}} \\ \text{Cost in } £ = \frac{2500}{175}$
4. 4
   Finally, we do the division and round the answer to 2 decimal places, because money always has two decimal places for cents or pence.

   $2500 \div 175 = 14.285714... \\ \text{Rounding to 2 d.p. gives } £14.29 \\ \text{The souvenir cost } £14.29$

Worked Example: Analyzing Student Hobbies

Solving a Real-World Venn Diagram Problem 🎮

1. 1
   First, let's draw the basic Venn diagram. We need a rectangle for our universal set, E, which has 40 students in total. Inside, we'll draw two overlapping circles for our sets, S (Spotify) and N (Netflix).

   [IMAGE_PLACEHOLDER_2: A blank Venn diagram with two overlapping circles labeled S and N inside a rectangle labeled E.]
2. 2
   The golden rule of Venn diagrams is to always start from the middle! The middle is the intersection, $S \cap N$. The problem tells us that 10 students use both services, so we put '10' in the overlapping section.

   $n(S \cap N) = 10$
3. 3
   Now, let's figure out the 'Spotify only' part. We know the entire Spotify circle, $n(S)$, must have 25 students. Since 10 are already in the overlap, the number of students who only use Spotify is the total minus the overlap.

   $\text{Spotify only} = n(S) - n(S \cap N) = 25 - 10 = 15$
4. 4
   We do the same thing for the 'Netflix only' part. The whole Netflix circle, $n(N)$, has 22 students. Subtract the 10 who are also in the Spotify group to find those who are exclusive to Netflix.

   $\text{Netflix only} = n(N) - n(S \cap N) = 22 - 10 = 12$
5. 5
   Finally, we need to find how many students use neither service. This is the number that goes outside the circles but still inside the box. We add up everyone inside the circles and subtract that total from the universal set, E.

   $\text{Total in circles} = 15 (S \text{ only}) + 10 (Both) + 12 (N \text{ only}) = 37 <br> \text{Neither} = n(E) - 37 = 40 - 37 = 3$

Worked Example: Calculator Use — Fractions, Time and Brackets

1. 1
   For (a), bracket the numerator — otherwise the calculator thinks you meant $12.5 - \tfrac{3.5}{1.5}$. Different universe.

   $\frac{12.5 - 3.5}{1.5} = \frac{9}{1.5} = 6$
2. 2
   For (b), subtract times in base 60. $20 - 45$ doesn't work, so borrow 1 hour (= 60 min): $11\!:\!20$ becomes $10\!:\!80$.

   $10\!:\!80 - 8\!:\!45 = 2\!:\!35$
3. 3
   Now convert $2$ h $35$ min to total minutes.

   $2 \times 60 + 35 = 120 + 35 = 155 \text{ minutes}$
4. 4
   For (c), bracket the expression under the root. Recognise this as Pythagoras — nice round answer incoming. 😎

   $\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17$

Worked Example: Compound Interest

1. 1
   Lock in the compound interest formula. $P$ = starting amount, $r$ = rate, $n$ = number of years.

   $A = P \left(1 + \frac{r}{100}\right)^n$
2. 2
   Sub in $P = 2000$, $r = 5$, $n = 3$. Don't round yet — keep the full decimal.

   $A = 2000 \times (1.05)^3 = 2000 \times 1.157625$
3. 3
   Evaluate and round to 2 d.p. (money = 2 d.p., always).

   $A = \$2315.25$
4. 4
   For (b), interest is just the extra money — final amount minus what you started with.

   $\text{Interest} = 2315.25 - 2000 = \$315.25$