Edexcel GCSE Mathematics (9-1)

What are we being asked to find?

We need to find the value of a number with an exponent (or power/index). The big number is the base (\( 2 \)) and the small number is the power (\( 4 \)).

Why we do this:

A power tells us how many times to multiply the base number by itself. So \( 2^4 \) means we need to write the number \( 2 \) four times, and multiply them all together.

What this tells us:

By multiplying \( 2 \) by itself four times, we reach \( 16 \). Make sure not to accidentally calculate \( 2 \times 4 = 8 \), which is a very common mistake!

What are we being asked to find?

We are asked to round a number so it only has three digits after the decimal point. To do this properly, we have to look at the next digit to decide whether to round up or keep it the same.

Why we do this:

We identify the 3rd decimal place (the target) and the 4th decimal place (the decider). If the decider is \( 5 \) or more, we round the target up by one. If it’s \( 4 \) or less, the target stays the same.

What this tells us:

The number rounds up to \( 7.265 \). Everything after the 3rd decimal place is dropped.

What are we being asked to find?

We need to “simplify” an algebraic expression. This means writing it in the shortest, neatest way possible by combining terms where we can.

Why we do this:

In algebra, order doesn’t matter when we multiply. We can group the numbers together and multiply them, and then attach the letters. We also drop the multiplication signs (\( \times \)) to make it neat.

What this tells us:

The simplified form is \( 56ef \). (Note: writing \( 56fe \) is also mathematically correct, but alphabetical order is standard practice!).

What are we being asked to find?

We are asked to “solve” the equation, which means we need to find the value of the letter \( x \). Currently, \( x \) is being divided by \( 5 \).

Why we do this:

To get \( x \) on its own, we need to do the opposite (inverse) of dividing by \( 5 \). The opposite is multiplying by \( 5 \). We must multiply both sides of the equation by \( 5 \) to keep it balanced. It’s often easier to change the mixed number (\( 2\frac{1}{2} \)) into a decimal (\( 2.5 \)) before we multiply.

What this tells us:

The value of \( x \) is \( 12.5 \). We can check this by substituting it back in: \( 12.5 \div 5 = 2.5 \), which is exactly right!

What are we being asked to find?

We need to change a fraction into a percentage. “Percent” literally means “out of 100”. So, we are trying to find an equivalent fraction that has \( 100 \) as its denominator (bottom number).

Why we do this:

To turn the denominator from \( 5 \) into \( 100 \), we need to multiply it by \( 20 \) (because \( 100 \div 5 = 20 \)). Whatever we do to the bottom of the fraction, we must do exactly the same to the top to keep its value the same.

What this tells us:

\( \frac{80}{100} \) means “80 out of 100”, which is exactly the definition of 80 percent.

What are we being asked to find?

We need to calculate a percentage of an amount without a calculator. A common strategy for finding percentages manually is to start by finding 10%.

Why we do this:

To find 10% of any number, we divide it by 10. Once we know what 10% is, we can just multiply that value by 6 to find out what 60% is.

What this tells us:

We found that 60% of 70 is 42. We can do a quick mental check: 50% is half of 70, which is 35. Our answer of 42 is slightly bigger than 35, which makes perfect sense since 60% is slightly more than 50%.

What are we being asked to find?

We need to determine the probability of spinning a B and an F, and then mark these probabilities visually on the given scales from \( 0 \) to \( 1 \).

Why we do this:

Probability is calculated as: \(\frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\).

Why we do this:

We need to see how many sections have the letter ‘F’ on them. Since there are no sections with ‘F’, it is impossible to land on it.

What this tells us:

An event that is equally likely to happen as not happen has a probability of \( 1/2 \). An impossible event has a probability of \( 0 \).

What are we being asked to find?

We need to figure out the actual cost of a single packet of sausages to see if Fahima’s claim of \( £2.30 \) is correct. We have a list of items bought, money given, and change received.

Why we do this:

If we know she handed over \( £10 \) and got \( 30\text{p} \) (which is \( £0.30 \)) back, the difference is the total amount she spent on all her groceries.

Why we do this:

We need to find out how much the bread rolls and ketchup cost together so we can subtract this from the total spent. This will leave us with the cost of just the sausages.

Why we do this:

By subtracting the cost of the bread and ketchup from the total amount spent, we get the total cost of the \( 3 \) packets of sausages. Then, we divide by \( 3 \) to find the cost of one packet.

What this tells us:

One packet costs \( £1.70 \). Fahima thought it was \( £2.30 \). So, Fahima is wrong. We must clearly write our conclusion to get the final mark.

Why we do this:

When multiplying fractions, the rule is simple: we multiply the top numbers (numerators) together, and we multiply the bottom numbers (denominators) together. Unlike adding or subtracting, we do not need a common denominator.

What this tells us:

The answer is \( \frac{15}{32} \). We should quickly check if it simplifies, but \( 15 \) and \( 32 \) share no common factors (factors of 15 are 1, 3, 5, 15; none of these divide into 32). So, this is our final answer.

Why we do this:

We cannot subtract fractions if the bottom numbers (denominators) are different. We need to find the Lowest Common Multiple (LCM) of \( 3 \) and \( 4 \), which is \( 12 \). Then we convert both fractions so they have \( 12 \) as the denominator.

Why we do this:

Now that the denominators are the same, we simply subtract the top numbers (numerators) and keep the bottom number exactly as it is.

What this tells us:

The answer is \( \frac{5}{12} \). The number \( 5 \) is prime, and doesn’t divide into \( 12 \), so this fraction is fully simplified.

What are we being asked to find?

We need to calculate Sean’s total pay for a \( 10 \)-hour shift. His pay is split into two parts: normal hours (the first \( 8 \) hours) and overtime hours (anything over \( 8 \) hours, paid at a higher rate).

Why we do this:

We must first figure out how many hours are paid at the normal rate (\( £12 \)) and how many hours are overtime. Then we calculate the pay for the normal hours.

Why we do this:

We need to figure out what his new hourly rate is for the overtime. The question says it’s \( 1\frac{1}{4} \) times his normal rate. So we need to calculate \( 1\frac{1}{4} \) of \( £12 \).

Why we do this:

Now we just add the two amounts together to find his total earnings for Monday.

What this tells us:

Sean earned \( £126 \) total on Monday. Breaking the problem down into “normal pay” and “overtime pay” helps prevent confusing the hours.

What are we being asked to find?

We need to form a ratio matching the format “eggs in two boxes : total number of eggs”. To do this, we need to calculate those two amounts first.

Why we do this:

Ratios work just like fractions. To simplify them, we divide both sides by the highest number that goes evenly into both parts.

What this tells us:

The simplest form is \( 1:10 \). Shortcut: Because every box contains the same number of eggs, comparing eggs in 2 boxes to eggs in 20 boxes is exactly the same ratio as comparing the boxes themselves: \( 2 \text{ boxes} : 20 \text{ boxes} \), which simplifies directly to \( 1 : 10 \)!

Why we do this:

We need to find the rule connecting the pattern number to the number of square tiles. Let’s count them:
Pattern 1 has \( 1 \) square (\( 1 \times 1 \)).
Pattern 2 has \( 4 \) squares (\( 2 \times 2 \)).
Pattern 3 has \( 9 \) squares (\( 3 \times 3 \)).
The number of squares is the pattern number squared!

Why we do this:

Now we find the rule for circular tiles.
Pattern 1 has \( 4 \) circles (1 on each of the 4 sides).
Pattern 2 has \( 8 \) circles (2 on each of the 4 sides).
Pattern 3 has \( 12 \) circles (3 on each of the 4 sides).
The number of circles is \( 4 \times \) the pattern number.

Why we do this:

Derek claims: Odd pattern number \(\rightarrow\) Odd number of square tiles. We know the number of square tiles is the pattern number multiplied by itself (squared). We need to state a mathematical fact about multiplying odd numbers to prove him right or wrong.

What this tells us:

By using the general property of numbers (odd \(\times\) odd = odd), we can definitively prove the statement without having to check every single odd number to infinity.

What are we being asked to find?

We need the probability of picking a blue pen. Probability is written as a fraction: \(\frac{\text{number of blue pens}}{\text{total number of pens}}\).

Why we do this:

We know there are \( 7 \) successful outcomes (blue pens) out of \( 17 \) possible outcomes (total pens).

What this tells us:

The fraction \( \frac{7}{17} \) cannot be simplified further because \( 7 \) and \( 17 \) are both prime numbers.

Why we do this:

We need to know the typical real-world height of an adult man in metres. This is general knowledge required for estimation questions. An average man is usually between \( 1.7 \text{m} \) and \( 1.8 \text{m} \).

Why we do this:

We need to visually compare the height of the man to the height of the tree in the diagram. We can imagine stacking the man on top of himself to see how many “men” tall the tree is.

Why we do this:

We multiply our estimated real height of the man by our visual scale factor to get the estimated real height of the tree.

What this tells us:

The tree is roughly \( 10 \) metres tall. Because this is an estimation question, a wide range of answers is accepted as long as the logic is sound. (Mark scheme accepts anything between \( 7.5\text{m} \) and \( 12\text{m} \)).

Why we do this:

To draw a pie chart, we must turn the student numbers into degrees. A full circle is \( 360^\circ \). First, we find the total number of students, then figure out what fraction of the total each language represents, and multiply by \( 360^\circ \).

Why we do this:

Now we plot these angles using a protractor, starting from the vertical line given, and label each sector.

Why we do this:

Shameena is comparing the number of students based purely on looking at the pie charts. We need to remember what a pie chart actually tells us.

What this tells us:

You cannot compare actual quantities between two pie charts unless you are given the total number of data points for both charts. Shameena is wrong.

Why we do this:

We are given information connecting the areas of the two shapes. We have all the measurements needed to find the area of the right-angled triangle, so we start there using the formula: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\).

Why we do this:

The problem states that the rectangle’s area is exactly \( 6 \) times larger than the triangle’s area. So we multiply our result by \( 6 \).

Why we do this:

The area of a rectangle is \(\text{length} \times \text{width}\). We know the area is \( 216 \) and the length is \( 16 \). To find the missing width, we divide the area by the length (\( 216 \div 16 \)). We can make this easier by simplifying it like a fraction.

What this tells us:

The width of the rectangle is \( 13.5 \text{ cm} \). We can check this by calculating \( 16 \times 13.5 \). \( 16 \times 10 = 160 \), \( 16 \times 3 = 48 \), \( 16 \times 0.5 = 8 \). \( 160 + 48 + 8 = 216 \). The math is perfect.

What are we being asked to find?

We need to substitute the given numerical values into the algebraic formula to calculate \( v \). Remember that \( at \) means \( a \times t \).

Why we do this:

We replace each letter with its corresponding number. Then, we must follow the order of operations (BIDMAS/BODMAS), doing the multiplication before the addition.

What this tells us:

The value of \( v \) is \( -0.5 \) (or \( -\frac{1}{2} \) if you kept it as a fraction). Either format is perfectly acceptable for full marks.

What are we being asked to find?

We need to figure out the individual weights of a “tin” and a “packet” so we can calculate a new combination. We start with the tins, because we are told that exactly \( 5 \) of them weigh \( 1750\text{g} \).

Why we do this:

We know that \( 4 \) tins + \( 3 \) packets = \( 1490\text{g} \). Now that we know the weight of a tin, we can calculate the weight of the \( 4 \) tins, subtract that from the total, and be left with just the weight of the \( 3 \) packets.

Why we do this:

The question asks for the weight of \( 3 \) tins and \( 2 \) packets. We simply multiply our individual weights by these amounts and add them together.

What this tells us:

The final combined weight is \( 1110 \) grams. By breaking the problem down into finding the weight of a single item first, complex word problems become much simpler.

What are we being asked to find?

We first need to estimate the area of the circular garden. The formula for the area of a circle is \( A = \pi \times r^2 \). Since this is an estimation question on a non-calculator paper, we should estimate the value of \( \pi \) (which is approx \( 3.14159… \)) as \( 3 \).

Why we do this:

We need to divide our total area by the coverage of one box to find out how many boxes are needed. To make the division easier without a calculator, we can round the box coverage from \( 46\text{ m}^2 \) to \( 50\text{ m}^2 \).

Why we do this:

To determine if our estimate is too low (underestimate) or too high (overestimate), we compare our rounded numbers to the true numbers.

What this tells us:

When you round the numerator down (Area) and the denominator up (Coverage), your final division result will always be smaller than the true mathematical answer.

Why we do this:

To find the value of \( x \), we must first deal with the brackets. We can either expand the brackets by multiplying everything inside by \( 4 \), or we can divide both sides of the equation by \( 4 \) to remove it. Let’s expand the brackets.

Why we do this:

An “integer” is a whole number (positive, negative, or zero). We need to list all the whole numbers that satisfy the inequality \( -3 < t \le 2 \).

• \( < \) means "strictly less than". So \( t \) must be bigger than \( -3 \), meaning it cannot be \( -3 \).
• \( \le \) means “less than or equal to”. So \( t \) can be smaller than \( 2 \) OR it can be exactly \( 2 \).

What this tells us:

These five numbers are the only integers that fit into the given range. Missing a number or including \( -3 \) are the most common mistakes here.

Why we do this:

To find out how much his new pay is, we first need to figure out exactly how much extra money the \( 3\% \) increase represents. Without a calculator, finding \( 1\% \) first is the easiest method.

Why we do this:

The question asks for “how much money Azmol will be paid”, which means the final total, not just the extra amount. We must add the increase to his original pay.

What this tells us:

Azmol’s new monthly salary is \( £1545 \). It’s a slightly higher number than his old salary, which perfectly aligns with a small \( 3\% \) increase.

What are we being asked to find?

An outlier is a point that does not fit the general pattern or trend of the rest of the data. We need to find its \((x, y)\) coordinates.

Why we do this:

Correlation describes the relationship between the two variables. We look at the general direction the points are heading. If they go up together, it’s positive. If one goes down while the other goes up, it’s negative.

Why we do this:

To estimate a value, we should ideally draw a Line of Best Fit (a straight line drawn through the middle of the points, following the trend). Then we find \( 16.4^\circ\text{C} \) on the y-axis, move across to the line, and read down to the x-axis.

Why we do this:

We need to see if the real data on the graph matches the weatherman’s claim. He is describing a positive correlation.

What this tells us:

Yes, the graph supports the statement because the correlation is positive.

What are we being asked to find?

We need to break down the number \( 56 \) by dividing it only by prime numbers (\( 2, 3, 5, 7, 11… \)) until we can’t divide any further. The final answer must be written as a multiplication (“product”).

Why we do this:

A factor tree helps us systematically split the number. Start by finding any two numbers that multiply to make \( 56 \), then keep breaking those down.

What this tells us:

The prime factors are \( 2, 2, 2 \), and \( 7 \). The question asks for the “product”, so we must write them multiplied together. You can also group the \( 2 \)’s using index notation.

Why we do this:

It’s much easier to multiply whole numbers. So we pretend the decimal points aren’t there and calculate \( 546 \times 43 \). Later, we will put the decimal point back into the final answer.

Why we do this:

We use the standard column method to multiply a 3-digit number by a 2-digit number.

Why we do this:

We need to adjust our whole-number answer to account for the original decimals. We count how many digits were behind the decimal point in the original problem, and place the point in our answer so it has the same amount.

What this tells us:

We can check with a quick estimation: \( 50 \times 4 = 200 \). Our answer is \( 234.78 \), which is close to \( 200 \), meaning our decimal point is absolutely in the right place.

What are we being asked to find?

We need to show how the geometric information given turns into the specific quadratic equation shown. This means creating an algebra equation for the area and rearranging it.

There are two ways to find the area of the whole square: treat it as one large square, or add up the \( 4 \) smaller inside sections.

Why we do this:

We are told the total area equals \( 10 \). We set our algebraic expression equal to \( 10 \) and expand it.

Why we do this:

The question wants us to show that the equation can look like \( x^2 + 6x = 1 \). We need to move the \( 9 \) to the other side of the equals sign to make it match.

What this tells us:

We successfully started with the geometric facts and manipulated the algebra to perfectly match the target equation. In “Show that” questions, the final line MUST be exactly what the question asked for.

What are we being asked to find?

We need the total length of all \( 5 \) pieces of metal before we can find the total weight. The frame has \( 2 \) lengths of \( 12\text{m} \), \( 2 \) widths of \( 5\text{m} \), but we are missing the diagonal length.

Why we do this:

Because the frame is a rectangle, the diagonal creates a right-angled triangle with sides \( 5 \) and \( 12 \). We can use Pythagoras’ Theorem (\( a^2 + b^2 = c^2 \)) to find the hypotenuse (\( c \)).

Why we do this:

Now we have all the pieces, we add them together to find the total length of metal used.

Why we do this:

Each single metre weighs \( 1.5\text{ kg} \). So we multiply our total length by \( 1.5 \) to find the overall weight.

What this tells us:

The total weight of the frame is \( 70.5\text{ kg} \). This type of multi-stage problem requires careful reading to ensure you don’t stop prematurely after finding the perimeter.

What are we being asked to find?

We need to prove that two lines run in exactly the same direction and never touch. Mathematically, this means we must show that they have exactly the same gradient (steepness).

Why we do this:

When the equation of a straight line is written in the standard form \( y = mx + c \), the number attached to the \( x \) (the letter \( m \)) is the gradient.

Why we do this:

\( L_2 \) is currently a jumbled equation. We cannot just look at the number in front of \( x \) right now, because it is not in the \( y = mx + c \) format. We must rearrange it to get a single \( y \) on its own on the left side.

What this tells us:

Both lines have exactly the same gradient (\( m = 3 \)), which is the mathematical proof that they are parallel. We must write a concluding sentence to finish the “show that” question properly.

Why we do this:

We need the vector journey starting at \( D \) and ending at \( B \). A key property of any parallelogram is that its diagonals cut exactly in half (bisect each other). This means the journey from \( D \) to \( O \) is identical to the journey from \( O \) to \( B \).

Why we do this:

We need the vector to travel directly from \( A \) to \( B \). Since we don’t have a direct vector for this path yet, we must travel along paths we *do* know: from \( A \) to \( O \), and then from \( O \) to \( B \).

Why we do this:

We need the journey from \( A \) to \( D \). We can find this by travelling from \( A \) to \( O \), and then from \( O \) to \( D \).

What this tells us:

Vector geometry is entirely about finding a path from the start point to the end point using the “building block” vectors you have been given. Reversing direction simply flips the sign from positive to negative.