Chapter 9 — Two Full Practice Papers

Answer: D

Explanation

$$240 \times 0.30 = 72$$

C gives the households that do not separate glass; B takes a third rather than 30%.

Takeaway: "of" means multiply. Check whether the question wants the group described or the rest.

Answer: G

Explanation

Undo the operations in the reverse of the order they were applied: the $+5$ first, then the multiplication by 3.

Subtract 5, then divide by 3:

$$3x = 15 \qquad x = 5$$

Takeaway: undo in reverse order — the addition first, then the multiplication.

Answer: D

Explanation

Find what happens in one unit of time first, then scale up. Going straight from 3 minutes to 8 tempts you into multiplying by 8 directly.

$45 \div 3 = 15$ bottles per minute, so $15 \times 8 = 120$.

Takeaway: find the rate for one unit first, then multiply.

Answer: F

Explanation

The three angles of any triangle total $180°$, so the third is whatever is left after the other two.

$$180 - 35 - 95 = 50°$$

Check: $35 + 95 + 50 = 180$.

Takeaway: subtract both known angles from $180°$, then add all three back.

Answer: B

Explanation

The number outside the bracket multiplies both terms inside it, signs included.

$$4 \times 2x = 8x \qquad 4 \times (-5) = -20$$

Takeaway: the number outside multiplies both terms inside.

Answer: G

Explanation

Multiplying powers of the same base adds the exponents — you are just counting how many copies of $x$ are being multiplied together.

Multiplying powers of the same base adds the exponents: $5 + 3 = 8$.

Takeaway: multiply → add exponents; divide → subtract; power of a power → multiply.

Answer: D

Explanation

Substitute 5 for every $x$, taking the power first:

$$f(5) = 5^2 - 3(5) = 25 - 15 = 10$$

Takeaway: apply the power before the subtraction.

Answer: J

Explanation

$4 + 5 = 9$ parts, so one part is $45 \div 9 = 5$. The larger share is $5 \times 5 = 25$.

Takeaway: add the parts first, then multiply by the part asked for.

Answer: D

Explanation

$$f(0) = 4(0) + 7 = 7$$

It is the handling fee — what you pay before travelling anywhere.

Takeaway: the constant term is the value at zero: the fixed part of a linear model.

Answer: F

Explanation

The mean is the total shared equally, so add everything and divide by how many there are.

$$\frac{8 + 11 + 6 + 14 + 11}{5} = \frac{50}{5} = 10$$

Takeaway: mean is total ÷ count. The mode (11) and the range (8) are different things.

Answer: B

Explanation

Substitute the first into the second:

$$3x + (2x - 1) = 14 \quad\Longrightarrow\quad 5x = 15 \quad\Longrightarrow\quad x = 3$$

C is $y$, which is 5.

Takeaway: when one equation already gives a variable alone, substitute. Then check which letter was asked for.

Answer: H

Explanation

A 15% rise multiplies by $1.15$ — the original whole, plus another 15% of it:

$$80 \times 1.15 = 92$$

Takeaway: increase by $p\%$ in one step, multiplying by $1 + \frac{p}{100}$.

Answer: D

Explanation

Work outwards from the inner bracket. $f(4) = 4 + 3 = 7$, and that 7 is then the input to $g$, giving $g(7) = 14$.

Takeaway: work from the inside bracket outwards.

Answer: G

Explanation

Area uses the square of the radius; circumference does not. Check which the question wants before choosing the formula.

$$A = \pi r^2 = 49\pi$$

F is the circumference, $2\pi r$.

Takeaway: area squares the radius; circumference does not.

Answer: C

Explanation

Move the decimal point until exactly one non-zero digit stands in front of it, then count how many places it moved. A number below 1 gives a negative power.

The point moves 4 places right to give 3.6, so the power is $-4$.

Takeaway: a number below 1 gets a negative power. Expand your answer to check.

Answer: H

Explanation

The multiples of 6 up to 30 are $6, 12, 18, 24, 30$ — five of them.

$$P = \frac{5}{30} = \frac{1}{6}$$

Takeaway: count the outcomes that work, then divide by how many there are altogether.

Answer: D

Explanation

Add $2x$ to both sides and add 15:

$$24 < 6x \quad\Longrightarrow\quad 4 < x$$

Strictly greater than 4, so the least integer is 5.

Takeaway: a strict inequality excludes its boundary. Gather $x$ on the side that keeps it positive and no sign flip is needed.

Answer: J

Explanation

$$\frac{30 \times 5280}{3600} = 44$$

Remember 60 mph = 88 ft/s; half the speed is half the answer.

Takeaway: write both conversions as fractions and let the units cancel.

Answer: B

Explanation

The strut runs corner to corner, so it is the hypotenuse of a right triangle whose legs are the two sides of the gate.

$$\sqrt{81 + 144} = \sqrt{225} = 15$$

A 9-12-15 triangle — three times 3-4-5.

Takeaway: square, add, then root. Recognising the triple saves the arithmetic.

Answer: H

Explanation

The $-5$ is outside the brackets, so it lowers every output by 5. The $x$-coordinate is unchanged.

Takeaway: outside the brackets moves the graph vertically, in the direction the sign says.

Answer: D

Explanation

Check first that the sequence is arithmetic by subtracting consecutive terms. They differ by 14 every time.

The common difference is 14, so

$$7 + 9 \times 14 = 133$$

Takeaway: $a + (n-1)d$ — nine steps to reach the tenth term.

Answer: G

Explanation

The four total $4 \times 21 = 84$; the three given total 61; so the fourth is 23.

Takeaway: mean × count = total. Get the total first.

Answer: A

Explanation

Two numbers multiplying to $-21$ and adding to $+4$: $+7$ and $-3$.

$$(x + 7)(x - 3) = 0 \quad\Longrightarrow\quad x = -7 \text{ or } x = 3$$

Time cannot be negative, so $x = 3$.

Takeaway: factor, set each bracket to zero, then reject any root the context forbids.

Answer: J

Explanation

A cylinder's volume is the area of its circular base multiplied by its height, so only the radius gets squared.

$$V = \pi r^2 h = \pi \times 9 \times 10 = 90\pi$$

Takeaway: only the radius is squared.

Answer: B

Explanation

Restricted to cyclists, so the denominator is 80:

$$\frac{30}{80} = \frac{3}{8}$$

Takeaway: the group named after "from those who" is the denominator.

Answer: F

Explanation

Set the two ratios equal with the same quantity on top of both, then scale. More biscuits must mean more flour, which rules out any smaller answer.

24 biscuits is three times 8, so the flour is three times 3, which is 9.

Takeaway: keep the same quantity on top of both fractions, and check the direction — more biscuits means more flour.

Answer: C

Explanation

A fraction has no value only when its denominator is zero. Setting the numerator to zero answers a different question — where the graph crosses the $x$-axis.

A fraction is undefined only when its denominator is zero: $x = 7$.

Takeaway: the bottom decides the domain; the top decides the zeros.

Answer: J

Explanation

$$m = \frac{-3 - 7}{3 - (-2)} = \frac{-10}{5} = -2$$

The line falls, so a negative slope is right.

Takeaway: subtract in the same order top and bottom, and watch the double negative.

Answer: A

Explanation

Isolate the term containing $w$ first, then divide by everything multiplying it. Test the result with easy numbers.

$$P - 2l = 2w \quad\Longrightarrow\quad w = \frac{P - 2l}{2}$$

Takeaway: isolate the term, then divide by everything multiplying it. Test with small numbers.

Answer: H

Explanation

Sort the values first — this is the step people skip. With an even count there is no single middle value, so average the middle two.

Sorted: $2, 4, 6, 8, 9, 11$. The middle two are 6 and 8, so the median is 7.

Takeaway: sort first. With an even count, average the middle two.

Answer: B

Explanation

No solution means the $x$ terms cancel and the constants disagree, so $k = 6$ and the equation becomes $5 = -4$, which is false.

Takeaway: no solution needs matching coefficients and differing constants; infinitely many needs both to match.

Answer: G

Explanation

$$\frac{h}{9} = \frac{2}{1.5} \quad\Longrightarrow\quad h = 9 \times \frac{4}{3} = 12$$

Takeaway: put the same measurement on top of both fractions, then sanity-check the size.

Answer: C

Explanation

Count the doubling periods, not the hours: 12 hours at one doubling every 3 hours is four doublings, not twelve.

$12 \div 3 = 4$ doublings, so $60 \times 2^4 = 960$.

Takeaway: divide the time by the period to count the doublings.

Answer: J

Explanation

Rewrite each square root of a negative as a multiple of $i$ before combining them. The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ does not survive a negative under the sign.

$$\sqrt{-9} = 3i \qquad \sqrt{-16} = 4i \qquad 3i + 4i = 7i$$

Takeaway: convert each root to a multiple of $i$ before combining. The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ fails for negatives.

Answer: A

Explanation

Class A's scores sit within 2 marks of the mean; class B's spread 25 either side. B is far more spread out.

Takeaway: equal means say nothing about spread. That is what standard deviation measures.

Answer: F

Explanation

Percentage changes multiply rather than add, so take a convenient starting value of 100 and apply each multiplier in turn.

$$1.25 \times 0.80 = 1.00$$

A multiplier of exactly 1 means no overall change.

Takeaway: successive percentage changes multiply. Start from 100.

Answer: C

Explanation

The ramp is the hypotenuse and the height is opposite the angle, so use sine: $\text{height} = 9\sin 12°$.

Takeaway: pick the ratio linking the side you have to the side you want. A leg is always shorter than the hypotenuse.

Answer: H

Explanation

Squaring gives $x^2 - x - 6 = 0$, so $x = 3$ or $x = -2$. Testing back, $\sqrt{4} = 2 \ne -2$, so $-2$ is extraneous.

Takeaway: after squaring, every root is only a candidate. Substitute back.

Answer: D

Explanation

A logarithm states which power the base must be raised to, so rewrite it as an ordinary power immediately.

$\log_3(x) = 4$ means $3^4 = x$, so $x = 81$.

Takeaway: a logarithm is an exponent. Rewrite it as a power immediately.

Answer: G

Explanation

Two sides with the known angle between them is the signal for the law of cosines. Pythagoras would assume a right angle, and this one is $60°$.

$$c^2 = 25 + 64 - 2(5)(8)(0.5) = 49 \quad\Longrightarrow\quad c = 7$$

Takeaway: two sides and the angle between them means the law of cosines.

Answer: A

Explanation

A scalar multiplies every entry, not just the first:

$$5\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 5 & 10 \\ 15 & 20 \end{pmatrix}$$

Takeaway: a scalar reaches all four entries. Count them.

Answer: F

Explanation

Times do not add — two pumps together must be faster than either alone, so the answer has to be under 6 hours. What adds is the rates.

Rates add: $\frac{1}{6} + \frac{1}{12} = \frac{1}{4}$, so 4 hours.

Takeaway: add the rates, then invert. The answer must be less than the faster worker alone.

Answer: B

Explanation

Each bread can go with each filling, and each of those pairs with each sauce, so the three counts multiply:

$$4 \times 3 \times 2 = 24$$

Takeaway: independent choices multiply.

Answer: H

Explanation

A sector is the same fraction of the circle's area as its angle is of a full turn. Area uses $\pi r^2$; arc length would use $2\pi r$.

A $60°$ sector is one sixth of the circle: $\frac{1}{6} \times 36\pi = 6\pi$.

Takeaway: a sector is $\frac{\theta}{360}$ of the whole. Area uses $r^2$; arc length does not.

Answer: C

Explanation

One solution means $b^2 - 4ac = 0$:

$$100 - 4c = 0 \quad\Longrightarrow\quad c = 25$$

Then $x^2 + 10x + 25 = (x+5)^2$, zero only at $x = -5$.

Takeaway: exactly one solution means the discriminant is zero.

Paper 2

45 questions · 50 minutes · calculator permitted · no formula sheet

Do this one only after you have worked through what Paper 1 showed you.

Answer: G

Explanation

"What percent of 250" makes 250 the whole, so it goes underneath:

$$\frac{40}{250} \times 100 = 16\%$$

Check the other way: 16% of 250 is 40.

Takeaway: the number after "percent of" goes underneath. Over 100% means you divided upside down.

Answer: D

Explanation

Either divide by the 2 first or expand the bracket fully. Doing half of each is what produces the wrong answers here.

Divide by 2 first: $x + 6 = 13$, so $x = 7$.

Takeaway: either divide by the bracket's multiplier first, or expand fully. Do not do half of each.

Answer: J

Explanation

Carpet covers a surface, so this is an area, not a perimeter. Then finish the question — the area is not the answer.

Area is $24$ m², so the cost is $24 \times 18 = 432$.

Takeaway: covering a floor is an area. Finish the question — the area is rarely the answer.

Answer: B

Explanation

The mode is the value that appears most often. Counting: 4 appears three times, 7 twice, and 9, 2 and 5 once each.

$$\text{mode} = 4$$

Takeaway: mode is the most frequent value, not the middle one and not the average. Tally before answering.

Answer: H

Explanation

$$10x - 15 - 2x - 8 = 8x - 23$$

The $-2$ multiplies both terms of the second bracket.

Takeaway: a minus in front of a bracket changes every sign inside it.

Answer: D

Explanation

"Greater than 4" means 5 or 6 — two outcomes out of six.

$$P = \frac{2}{6} = \frac{1}{3}$$

Note that 4 itself is not included; "greater than" excludes the number named.

Takeaway: count the outcomes that qualify, then divide by the total. Read whether the boundary value counts.

Answer: F

Explanation

The whole numerator is divided by 2, not just part of it, so work out the top before dividing.

$$g(-4) = \frac{-4 + 6}{2} = \frac{2}{2} = 1$$

Takeaway: the whole numerator is divided, not just part of it.

Answer: C

Explanation

Convert the time into the unit the rate uses before multiplying. The rate is per minute, so the 2 hours must become minutes.

2 hours is 120 minutes, so $3 \times 120 = 360$ litres.

Takeaway: convert the time to the rate's unit before multiplying.

Answer: H

Explanation

The range is the largest value minus the smallest, and nothing else:

$$12 - 5 = 7$$

Here the mode is 5 and the mean is about 7.2 — three different numbers from the same six values.

Takeaway: range is largest minus smallest. The mode here is 5 and the mean is about 7.2 — three different numbers.

Answer: A

Explanation

Here 63 kg is one part of the ratio, not the whole. Find what a single part is worth, then multiply by the number of parts wanted.

One part is $63 \div 7 = 9$ kg, so cement is $2 \times 9 = 18$ kg.

Takeaway: here the 63 is one part, not the whole. Read which quantity the number belongs to.

Answer: J

Explanation

Subtracting the second equation from the first removes $x$:

$$4y = 12 \quad\Longrightarrow\quad y = 3$$

Takeaway: matching coefficients means subtract. Then check which letter was asked for.

Answer: C

Explanation

After a 20% reduction the customer pays 80% of the original, so undo it by dividing. Adding 20% back on does not return you to the start.

$96$ is 80% of the original, so the original is $96 \div 0.8 = 120$.

Takeaway: to undo a percentage change, divide by the multiplier. Adding 20% back does not return you to the start.

Answer: F

Explanation

Horizontal shifts go inside the brackets and act opposite to their sign. Right by 3 means $(x - 3)^2$, whose lowest point is at $x = 3$.

Takeaway: inside and opposite for horizontal; outside and obvious for vertical.

Answer: B

Explanation

$(x - h)^2 + (y - k)^2 = r^2$ with $h = 3$, $k = -2$, $r = 5$. The $-2$ becomes $(y + 2)$ and the right side is $25$.

Takeaway: the signs flip and the right-hand side is $r^2$.

Answer: H

Explanation

Averages of unequal groups cannot themselves be averaged. Convert each class to a total, add the totals, and divide by the combined count.

$$\frac{20(68) + 30(78)}{50} = \frac{3700}{50} = 74$$

Takeaway: averages of unequal groups cannot be averaged. Go back to totals.

Answer: D

Explanation

Subtracting consecutive terms gives no constant, so try dividing: each term is 4 times the one before. That makes it geometric.

The ratio is 4, so the 6th term is $3 \times 4^5 = 3072$.

Takeaway: $a \cdot r^{\,n-1}$ — the exponent is one less than the term number.

Answer: G

Explanation

Total drinks:

$$1(5) + 2(10) + 3(5) = 5 + 20 + 15 = 40$$

Customers: $5 + 10 + 5 = 20$.

$$\text{mean} = \frac{40}{20} = 2$$

Takeaway: multiply each value by its frequency, add, then divide by the total frequency. Averaging the top row alone ignores how many people gave each answer.

Answer: A

Explanation

The multiplier is what the machine keeps, so what it loses is the rest. Subtract from 1 rather than reading $0.85$ as the loss.

A multiplier of $0.85$ keeps 85%, so 15% is lost each year.

Takeaway: the multiplier is what is kept. Subtract from 1 for what is lost.

Answer: J

Explanation

Without replacement, both the count of reds and the total drop by one before the second draw.

$$\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$$

Takeaway: without replacement, both the top and the bottom drop by one on the second draw.

Answer: A

Explanation

Dividing an inequality by a negative number reverses its direction. Test one value from your answer to confirm.

$-3x > 15$, and dividing by $-3$ reverses the sign: $x < -5$.

Takeaway: dividing by a negative flips the inequality. Test one value to confirm.

Answer: F

Explanation

Not red means green or yellow: $3 + 5 = 8$ counters out of 20.

$$\frac{8}{20} = \frac{2}{5}$$

Or use the complement: red is $\frac{12}{20} = \frac{3}{5}$, so not red is $1 - \frac{3}{5} = \frac{2}{5}$.

Takeaway: the denominator is everything possible. For a "not" question, $1 - P(A)$ is usually the quicker route.

Answer: C

Explanation

The vertex of $y = ax^2 + bx + c$ sits at $x = -\frac{b}{2a}$, and $b$ carries its own sign — here two minus signs cancel.

$$x = -\frac{b}{2a} = -\frac{-6}{2} = 3$$

Takeaway: $b$ carries its own sign, and the formula has a minus of its own. Two minuses make a plus.

Answer: H

Explanation

Multiply by $\frac{\pi}{180}$. The $\pi$ stays in the answer; if it has vanished, the conversion went the wrong way.

$$135 \times \frac{\pi}{180} = \frac{3\pi}{4}$$

Takeaway: multiply by $\frac{\pi}{180}$. The $\pi$ stays in the answer.

Answer: B

Explanation

The determinant is the product along the main diagonal minus the product along the other one:

$$ad - bc = (4)(5) - (3)(2) = 20 - 6 = 14$$

Takeaway: main diagonal first, then subtract the other.

Answer: G

Explanation

The 12 who play both have been counted twice, once in each total, so subtract them once:

$$34 + 28 - 12 = 50$$

Check by parts: 22 chess only, 16 draughts only, 12 both.

Takeaway: subtract the overlap once — it was counted twice.

Answer: C

Explanation

Slope is the change in output divided by the change in input, and on a linear table it is the same between any two rows.

$$\frac{21 - 9}{4 - 1} = \frac{12}{3} = 4$$

Takeaway: slope is change in output over change in input, and on a linear table it is the same between any two rows.

Answer: H

Explanation

A perfect square is $(x + a)^2 = x^2 + 2ax + a^2$. Here $a^2 = 36$ so $a = 6$, and the middle term is $2a = 12$.

Takeaway: in a perfect square the middle coefficient is twice the square root of the constant.

Answer: A

Explanation

Do the same thing to all three parts at once, and copy each inequality sign down exactly as it was — the ends decide between the options here.

Subtract 1 from all three parts and divide by 2:

$$-4 < x \le 4$$

Takeaway: operate on all three parts, and carry each inequality sign down unchanged.

Answer: J

Explanation

With $a + c = 60$ and $14a + 9c = 710$, substituting $a = 60 - c$ gives $840 - 5c = 710$, so $c = 26$ and $a = 34$.

Takeaway: name both unknowns, write one equation for the count and one for the money.

Answer: A

Explanation

$$25 + 4n = 65 \quad\Longrightarrow\quad 4n = 40 \quad\Longrightarrow\quad n = 10$$

Takeaway: set the two totals equal. Remove the fixed part before dividing.

Answer: F

Explanation

The first line has slope 2. The second has slope $-\frac{6}{k}$, so $-\frac{6}{k} = 2$ gives $k = -3$. The intercepts then differ, so the lines are parallel and distinct.