Chapter 1 — Integrating Essential Skills

Answer: D

Explanation

The rate is 3 pages per 4 seconds, so per second it is

$$\frac{3}{4} = 0.75 \text{ pages per second}$$

One minute is 60 seconds, so

$$0.75 \times 60 = 45$$

Check the units: $\dfrac{\text{pages}}{\text{second}} \times \text{seconds}$ leaves pages. Correct.

Why each wrong option is wrong:

• B, $80$ — used $\frac{4}{3}$ instead of $\frac{3}{4}$, that is, 4 pages per 3 seconds. Turning a rate upside down is the most common slip in this topic. Ask yourself which number is bigger and whether the answer should be.
• C, $12$ — multiplied 3 by 4. Those are the two halves of one rate, not two quantities to combine.
• A, $180$ — read the rate as 3 pages per second and ignored the 4.

Takeaway: write the rate as a fraction with units attached, then multiply so the unwanted unit cancels. If the surviving unit is wrong, so is the arithmetic.

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Answer: G

Explanation

Change both units in one chain, and let them cancel:

$$30 \; \frac{\text{miles}}{\text{hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}$$

Miles cancel against miles, hours against hours, and feet over seconds survives — which is what was asked for.

$$\frac{30 \times 5280}{3600} = \frac{158400}{3600} = 44$$

A sanity check worth remembering: 60 mph is 88 feet per second. So 30 mph should be half of that, 44. It is.

Why each wrong option is wrong:

• F, $\frac{225}{11}$ — multiplied by 3600 and divided by 5280, the two conversions the wrong way round.
• H, $5280$ — converted correctly to feet per minute and stopped there.
• J, $\frac{22}{15}$ — converted 1 mile per hour instead of 30.

Takeaway: when both units change, write both conversions as fractions in one line and cancel. And keep "60 mph = 88 ft/s" in your head as a check.

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Answer: D

Explanation

Let $h$ be the number of hours. The bill is the fixed part plus the hourly part:

$$45 + 28h = 171$$

Take the call-out fee off first, because it is charged once and is not part of the rate:

$$28h = 126$$

$$h = \frac{126}{28} = \frac{9}{2} = 4.5 \text{ hours}$$

Check: $28 \times 4.5 = 126$, and $126 + 45 = 171$. Correct.

Why each wrong option is wrong:

• A, $6$ — subtracted the fee correctly but then divided by 21 instead of 28.
• B, $\frac{171}{28}$ — divided the whole bill by the hourly rate, as though there were no call-out fee. This is the trap the question is built around.
• C, $\frac{54}{7}$ — added the call-out fee instead of subtracting it.

Takeaway: a fixed charge is not part of the rate. Remove it before you divide, never after.

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Answer: J

Explanation

"Value" here means grams per dollar, so divide grams by dollars for each:

$$\text{250 g bag: } \frac{250}{6} = 41\tfrac{2}{3} \text{ g per dollar}$$

$$\text{400 g bag: } \frac{400}{9} = 44\tfrac{4}{9} \text{ g per dollar}$$

The larger bag is better value. The question asks for the difference:

$$\frac{400}{9} - \frac{250}{6} = \frac{800}{18} - \frac{750}{18} = \frac{50}{18} = \frac{25}{9}$$

Why each wrong option is wrong:

• F, $\frac{125}{3}$ and G, $\frac{400}{9}$ — each gives one bag's rate rather than the difference between them. Both are steps on the way; neither is the answer.
• H, $150$ — compared the weights, $400 - 250$, ignoring price altogether.

Takeaway: "better value" always means a rate, never a raw size or a raw price. And read the last line: this question wanted the difference, not the winner.

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Answer: B

Explanation

Convert first:

$$5 \text{ gallons} \times 16 \frac{\text{cups}}{\text{gallon}} = 80 \text{ cups}$$

The rate is 2 scoops per 4 cups, which is $\frac{2}{4} = \frac{1}{2}$ scoop per cup:

$$80 \times \frac{1}{2} = 40 \text{ scoops}$$

Why each wrong option is wrong:

• A, $160$ — multiplied the 80 cups by 2 instead of by $\frac{2}{4}$.
• C, $\frac{5}{2}$ — applied the rate to 5 gallons without converting to cups. The unit check catches this instantly: the answer would be in scoops per gallon, not scoops.
• D, $20$ — divided by 4 but never multiplied by the 2.

Takeaway: convert to a common unit before applying a rate, not after.

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Answer: G

Explanation

Go one step at a time, and never skip a unit:

$$3 \frac{\text{litres}}{\text{minute}} \times 60 \frac{\text{minutes}}{\text{hour}} = 180 \frac{\text{litres}}{\text{hour}}$$

$$180 \frac{\text{litres}}{\text{hour}} \times 24 \frac{\text{hours}}{\text{day}} = 4320 \frac{\text{litres}}{\text{day}}$$

Why each wrong option is wrong:

• F, $180$ — stopped after the first conversion, giving litres per hour. Stopping one step early is the single most common error in this whole chapter.
• H, $72$ — jumped from minutes straight to days by multiplying by 24, which skips the factor of 60.
• J, $30240$ — went one conversion too far and gave litres per week.

Takeaway: do multi-step conversions one unit at a time and write each one down. The two failure modes are stopping early and going one step too far, and both are invisible unless the units are on the page.

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Answer: B

Explanation

Distance is speed multiplied by time, but only when the units agree. The speed is per hour, so convert 40 minutes to hours:

$$40 \text{ minutes} = \frac{40}{60} = \frac{2}{3} \text{ hour}$$

$$18 \times \frac{2}{3} = 12 \text{ km}$$

A quick sanity check: in a full hour the cyclist would do 18 km, and 40 minutes is two thirds of an hour, so the answer must be less than 18. It is.

Why each wrong option is wrong:

• A, $720$ — multiplied 18 by 40 directly, mixing hours with minutes. The result, 720 km in 40 minutes, fails the sanity check by a mile.
• C, $\frac{9}{20}$ — divided instead of multiplying.
• D, $27$ — used $\frac{60}{40}$ instead of $\frac{40}{60}$, turning the time fraction upside down, which makes the answer bigger than 18.

Takeaway: before multiplying a speed by a time, check they use the same unit of time. Then check the answer is on the right side of "one whole hour".

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Answer: G

Explanation

The mistake this question exists to catch is adding the times. Times do not add — think about it: two pumps together must be faster than either alone, so the answer has to be less than 6. That single check eliminates three of the four options before any arithmetic.

What does add is the rates. In one hour:

$$\text{pump 1 empties } \frac{1}{6} \text{ of the tank} \qquad \text{pump 2 empties } \frac{1}{12}$$

$$\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$$

Together they empty a quarter of the tank per hour, so the whole tank takes

$$1 \div \frac{1}{4} = 4 \text{ hours}$$

Why each wrong option is wrong:

• F, $18$ — added the two times. Two pumps cannot be slower than one.
• H, $9$ — averaged the two times, which is the same error in disguise: the answer is still not less than 6.
• J, $72$ — multiplied the times.

Takeaway: for combined-work questions, add the rates ($\frac{1}{t}$), then invert at the end. And check the answer is smaller than the fastest worker alone — it always must be.

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Section 2 — Percentages

A percentage is a fraction with 100 on the bottom. "17%" means $\frac{17}{100}$, or $0.17$. Every percentage question on the ACT is one of four shapes:

the question	what to do
what is $p\%$ of $N$?	multiply: $N \times \frac{p}{100}$
$A$ is what percent of $N$?	divide, then multiply by 100: $\frac{A}{N} \times 100$
increase $N$ by $p\%$	multiply by $\left(1 + \frac{p}{100}\right)$
decrease $N$ by $p\%$	multiply by $\left(1 - \frac{p}{100}\right)$

The two rows that cost marks are the last two. Increasing by 15% is multiplying by 1.15, in one step — not "find 15%, then add it", which is the same thing but twice as many chances to slip. And the reverse of increasing by 15% is dividing by 1.15, not decreasing by 15%.

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Answer: D

Explanation

Increasing by 15% means multiplying by $1.15$:

$$80 \times 1.15 = 92$$

If you would rather do it in two steps: 15% of 80 is $80 \times 0.15 = 12$, and $80 + 12 = 92$. Same answer, more steps, more chances to go wrong.

Why each wrong option is wrong:

• A, $12$ — gave the increase itself, $12$, and stopped. The question asked for the new price.
• B, $68$ — subtracted the 15% instead of adding it.
• C, $95$ — added 15 dollars. A percentage is a proportion of something, never a fixed amount.

Takeaway: increase by $p\%$ in one multiplication, by $\left(1 + \frac{p}{100}\right)$. Then check you answered the question that was asked — the increase, or the new total?

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Answer: J

Explanation

After a 20% reduction the customer pays 80% of the original. Call the original price $P$:

$$0.80 \times P = 96$$

$$P = \frac{96}{0.80} = 120$$

Check: 20% of 120 is 24, and $120 - 24 = 96$. Correct.

The trap here is worth dwelling on. Adding 20% back onto 96 gives $96 \times 1.2 = 115.20$, not 120. Taking 20% off and then putting 20% back on does not return you to where you started, because the second percentage is taken from a smaller number. To undo a multiplication, you divide.

Why each wrong option is wrong:

• F, $116$ and H, $76$ — treated the 20% as 20 dollars.
• G, $\frac{576}{5}$ — added 20% back on instead of dividing by 0.80. This is the intended trap, and it is very close to right.

Takeaway: to undo a percentage change, divide by the multiplier. Off by 20% means divide by 0.8; up by 20% means divide by 1.2.

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Answer: B

Explanation

Percentages do not add. Each change is a multiplication, so two changes one after the other multiply together.

Take a convenient starting price of 100:

$$100 \times 1.25 = 125 \qquad \text{then} \qquad 125 \times 0.80 = 100$$

Back exactly where it started. In one line:

$$1.25 \times 0.80 = 1.00$$

A multiplier of exactly 1 means no overall change.

The reason it works out so neatly: a 20% fall undoes a 25% rise because $\frac{1}{1.25} = 0.8$. They are inverses.

Why each wrong option is wrong:

• A, risen by 5% and C, fallen by 5% — subtracted the percentages, in one direction or the other. $25 - 20$ is not how successive changes combine.
• D, risen by 45% — added them.

Takeaway: successive percentage changes multiply. Start from 100, apply each multiplier in turn, and read the answer off directly.

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Answer: F

Explanation

"Of those bicycle owners" is the key phrase: the 25% applies to the 60%, not to the town.

Take a town of 100 people:

$$\text{bicycle owners} = 60 \qquad \text{daily riders} = 25\% \text{ of } 60 = 15$$

15 out of 100 is 15%.

In one line, "of" means multiply:

$$0.60 \times 0.25 = 0.15 = 15\%$$

Why each wrong option is wrong:

• G, $85$ — added the percentages.
• H, $35$ — subtracted them.
• J, $25$ — applied the 25% to the whole town, ignoring the fact that only bicycle owners were being described.

Takeaway: in percentage questions the word "of" means multiply. A percentage of a percentage is always smaller than either one.

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Answer: D

Explanation

A 35% profit on the cost means the selling price is 135% of the cost:

$$60 \times 1.35 = 81$$

In two steps: 35% of 60 is $60 \times 0.35 = 21$, and $60 + 21 = 81$.

Why each wrong option is wrong:

• A, $21$ — gave the profit, $21$, not the selling price.
• B, $95$ — added 35 dollars rather than 35 percent.
• C, $39$ — subtracted the profit, which would be selling at a loss.

Takeaway: profit "on cost" means multiply the cost by $\left(1 + \frac{\text{profit }\%}{100}\right)$. Then read the last line again: profit, or selling price?

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Answer: J

Explanation

"What percent of 250" tells you 250 is the whole, so it goes on the bottom:

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

Check it the other way: 16% of 250 is $250 \times 0.16 = 40$. Correct.

Why each wrong option is wrong:

• F, $625$ — divided 250 by 40, giving 625%. A part cannot be more than 100% of its whole, so this fails an instant sanity check.
• G, $40$ — repeated the 40 with a percent sign attached.
• H, $84$ — gave the percentage that was not set aside.

Takeaway: the number after "percent of" is the whole and goes underneath. If your answer is over 100%, you have divided upside down.

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Answer: C

Explanation

Discounts do not add, because the second is taken from the reduced price.

Start from 100. After 20% off you pay 80% of it; after a further 10% off you pay 90% of that:

$$100 \times 0.80 \times 0.90 = 72$$

So the customer pays 72% of the original, which means the reduction is

$$100 - 72 = 28\%$$

Why each wrong option is wrong:

• B, $30$ — added the two discounts. If that were right, ten successive 10% discounts would make everything free.
• A, $72$ — gave the percentage paid, not the percentage taken off. A very near miss, and the reason the last line of the question says "reduction".
• D, $2$ — multiplied 20 by 10 and divided by 100.

Takeaway: chain the multipliers, then subtract from 100 if the question asks for the discount rather than the price.

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Answer: F

Explanation

The same dollar amount each year, not the same percentage — so this is addition, not multiplication.

From year 1 to year 4 there are three rises, not four. That off-by-one is the whole question:

$$43000 - 34000 = 9000 \text{ over three rises}$$

$$\text{each rise} = \frac{9000}{3} = 3000$$

So year 2 is $34000 + 3000 = 37000$.

Check the whole run: 34000, 37000, 40000, 43000. Four years, three steps. Correct.

Why each wrong option is wrong:

• H, $36250$ — divided the 9000 by four instead of three: one for each year rather than one for each rise. This is the error the question is built around, and it is very easy to make quickly.
• G, $40000$ — subtracted one rise from the fourth year, which gives year 3.
• J, $43000$ — added the entire 9000 in one go.

Takeaway: between year 1 and year $n$ there are $n - 1$ steps. Count the gaps, not the years. Write the whole sequence out if there is any doubt — it takes five seconds and removes the error entirely.

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Section 3 — Ratios and Proportion

A ratio compares parts of the same whole. "The ratio of red to blue is $4 : 7$" does not mean 4 red and 7 blue; it means that for every 4 red there are 7 blue, in some multiple.

The reliable method is the multiplier. Write the ratio as parts, add them to get the total number of parts, and find what one part is worth:

$$4 : 7 \quad\longrightarrow\quad 4 + 7 = 11 \text{ parts}$$

If there are 44 objects altogether, one part is $44 \div 11 = 4$, so red is $4 \times 4 = 16$ and blue is $7 \times 4 = 28$.

The single most common error is using 4 out of 7 instead of 4 out of 11 — confusing a part-to-part ratio with a part-to-whole fraction.

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Answer: B

Explanation

Add the parts to find how many the whole is divided into:

$$4 + 7 = 11 \text{ parts}$$

$$\text{one part} = \frac{44}{11} = 4 \text{ marbles}$$

Red is 4 parts, so

$$4 \times 4 = 16$$

Check: blue is $7 \times 4 = 28$, and $16 + 28 = 44$. Correct.

Why each wrong option is wrong:

• A, $\frac{176}{7}$ — used $\frac{4}{7}$, treating a part-to-part ratio as a fraction of the whole. This is the error the question is built around.
• C, $28$ — worked correctly and gave the blue marbles.
• D, $4$ — gave the value of one part and stopped.

Takeaway: add the parts first. $4 : 7$ means $\frac{4}{11}$ of the whole, not $\frac{4}{7}$. Then check your two answers add back to the total.

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Answer: J

Explanation

$$5 + 3 + 2 = 10 \text{ parts}$$

$$\text{one part} = \frac{120}{10} = 12 \text{ cm}$$

The longest piece is 5 parts:

$$5 \times 12 = 60 \text{ cm}$$

Check: $60 + 36 + 24 = 120$. Correct.

Why each wrong option is wrong:

• F, $12$ — gave the size of one part.
• G, $24$ — gave the shortest piece, 2 parts.
• H, $36$ — gave the middle piece, 3 parts.

Takeaway: the method does not change with more parts — add them all, divide, then multiply by the part you were asked for. Read which one that is.

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Answer: C

Explanation

Set the two ratios equal, keeping flour on top in both:

$$\frac{3 \text{ cups}}{8 \text{ biscuits}} = \frac{c}{20 \text{ biscuits}}$$

Cross-multiply:

$$8c = 60 \qquad c = \frac{60}{8} = \frac{15}{2} = 7.5 \text{ cups}$$

Sanity check: 20 biscuits is two and a half times 8, so the flour should be two and a half times 3, which is 7.5. Correct.

Why each wrong option is wrong:

• A, $\frac{160}{3}$ — set the proportion up with biscuits over cups on one side and cups over biscuits on the other.
• B, $60$ — multiplied $3 \times 20$ and never divided by 8.
• D, $\frac{6}{5}$ — multiplied by 8 and divided by 20, the wrong way round.

Takeaway: in a proportion, put the same quantity on top of both fractions. Then check the direction: more biscuits must mean more flour.

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Answer: G

Explanation

Every member gets a whole number of tickets and a whole number of vouchers, so the number of members must divide both 30 and 75. The largest such number is the greatest common factor.

$$30 = 2 \times 3 \times 5 \qquad 75 = 3 \times 5 \times 5$$

The factors they share are $3 \times 5 = 15$.

Check: $30 \div 15 = 2$ tickets each and $75 \div 15 = 5$ vouchers each, both whole. Correct.

Why each wrong option is wrong:

• F, $5$ — 5 divides both, but it is not the largest number that does.
• H, $30$ — 30 does not divide 75.
• J, $150$ — gave the lowest common multiple, 150. A multiple is bigger than both numbers; the number of members has to be smaller.

Takeaway: "shared out with none left over, as many groups as possible" is the greatest common factor. "Happens again at the same time" is the lowest common multiple. Decide which before you calculate.

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Answer: D

Explanation

Each centimetre on the map stands for 25 real kilometres, so

$$7.5 \times 25 = 187.5 \text{ km}$$

The direction matters: going from the map to the world, the number gets bigger, so you multiply.

Why each wrong option is wrong:

• B, $\frac{3}{10}$ — divided by the scale, which is the conversion the other way round, from the world to the map.
• C, $25$ — repeated the scale factor.
• A, $32$ — added the two numbers.

Takeaway: decide which way the conversion goes before touching a calculator. Map to world, the number grows; world to map, it shrinks.

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Answer: F

Explanation

Let $n$ be the total. Apple pies take a third, so what remains is

$$n - \frac{1}{3}n = \frac{2}{3}n$$

Cherry pies are a quarter of that:

$$\frac{1}{4} \times \frac{2}{3}n = \frac{1}{6}n = 60$$

$$n = 360$$

Check: 360 pies, 120 apple, 240 remaining, and a quarter of 240 is 60. Correct.

Why each wrong option is wrong:

• G, $240$ — took the quarter from the whole day's sales, forgetting that the apple pies had already been removed.
• H, $720$ — divided by both fractions separately instead of by their product.
• J, $180$ — used the third rather than the quarter.

Takeaway: "of the remaining" means the fraction applies to what is left, not to the original. Write the leftover as a fraction of the total first.

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Section 4 — Area, Perimeter and Volume in Context

These questions are not really geometry. They are arithmetic wearing a shape: work out an area or a volume, then multiply by a cost or divide into a supply.

The ACT gives you no formula sheet, so the formulas themselves have to be known. The ones this section needs:

shape	formula
rectangle	area $= \text{length} \times \text{width}$
rectangle	perimeter $= 2(\text{length} + \text{width})$
triangle	area $= \tfrac{1}{2} \times \text{base} \times \text{height}$
circle	area $= \pi r^2$, circumference $= 2\pi r$
box	volume $= \text{length} \times \text{width} \times \text{height}$
cylinder	volume $= \pi r^2 h$

Chapter 8 has the complete list, with the reason each one is true. What matters here is the second half of the question: the units. An area in square metres multiplied by a cost per square metre gives dollars; anything else means something has gone wrong.

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Answer: C

Explanation

Area first:

$$6 \times 4 = 24 \text{ square metres}$$

Then the cost:

$$24 \times 18 = 432$$

Check the units: square metres $\times$ dollars per square metre gives dollars. Correct.

Why each wrong option is wrong:

• A, $24$ — gave the area, 24 square metres, and stopped. Stopping one step early is the commonest error in this chapter.
• B, $360$ — used the perimeter, $2(6 + 4) = 20$. Carpet covers a surface, so it is an area; perimeter would be for skirting board.
• D, $42$ — added the cost rather than multiplying by it.

Takeaway: covering a floor is area; going round an edge is perimeter. Then finish the question — the area is rarely the answer.

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Answer: G

Explanation

Volume of a box is length $\times$ width $\times$ height:

$$2 \times 1.5 \times 0.4 = 1.2 \text{ cubic metres}$$

Then convert:

$$1.2 \times 1000 = 1200 \text{ litres}$$

Why each wrong option is wrong:

• F, $\frac{6}{5}$ — gave the volume in cubic metres without converting to litres.
• H, $3900$ — added the three dimensions, $2 + 1.5 + 0.4 = 3.9$, before converting. Volume multiplies lengths; it never adds them.
• J, $12$ — the right shape of answer with a decimal point in the wrong place. Multiplying by 0.4 makes a number smaller, not ten times smaller.

Takeaway: volume multiplies three lengths. Then read the unit the answer is wanted in — it is often not the unit the measurements came in.

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Answer: A

Explanation

Find the missing side from the area:

$$5 \times w = 45 \qquad w = \frac{45}{5} = 9 \text{ m}$$

Now the perimeter, which is twice the sum of the two different sides:

$$2 \times (5 + 9) = 28 \text{ m}$$

Why each wrong option is wrong:

• B, $9$ — found the missing side, 9, and stopped there.
• C, $50$ — added an area to a length, which is not a meaningful operation — the units alone rule it out.
• D, $90$ — doubled the area.

Takeaway: an area divided by one side gives the other side. Watch the units: you cannot add square metres to metres.

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Answer: F

Explanation

Every measurement must be in the same unit before multiplying. The depth is 5 centimetres, and there are 100 centimetres in a metre:

$$5 \text{ cm} = \frac{5}{100} = 0.05 \text{ m}$$

$$12 \times 3 \times 0.05 = 1.8 \text{ cubic metres}$$

Why each wrong option is wrong:

• G, $180$ — multiplied by 5 without converting, treating the depth as 5 metres. A driveway five metres deep is a swimming pool; a sanity check catches this.
• H, $36$ — gave the area of the driveway, forgetting the depth entirely.
• J, $\frac{36}{5}$ — divided by the depth instead of multiplying.

Takeaway: convert every length to the same unit before you multiply, and then ask whether the answer is a sensible size for the thing described.

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Section 5 — Averages and Medians

The mean is the total divided by how many there are. Rearranged, that gives the fact almost every ACT average question turns on:

$$\text{total} = \text{mean} \times \text{how many}$$

Knowing the mean and the count gives you the total, and the total is what lets you recover a missing value or add a new one.

The median is the middle value once the data is in order. With an even number of values there is no single middle, so it is the mean of the middle two. Forgetting to sort first is the most common median error there is.

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Answer: C

Explanation

Five numbers with a mean of 20 must total

$$5 \times 20 = 100$$

The four known values add to

$$12 + 22 + 15 + 25 = 74$$

So the fifth is

$$100 - 74 = 26$$

Check: $12 + 22 + 15 + 25 + 26 = 100$, and $100 \div 5 = 20$. Correct.

Why each wrong option is wrong:

• A, $74$ — gave the total of the four known values.
• B, $20$ — assumed the missing value must equal the mean. It only does when the others happen to balance exactly, which is not the case here.
• D, $100$ — gave the total of all five.

Takeaway: mean $\times$ count $=$ total. Get the total first, then everything else is subtraction.

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Answer: J

Explanation

Sort them first — this is the step people skip:

$$2, \; 4, \; 6, \; 8, \; 9, \; 11$$

Six values, so there is no single middle one. The median is the mean of the third and fourth:

$$\frac{6 + 8}{2} = 7$$

Why each wrong option is wrong:

• F, $\frac{20}{3}$ — gave the mean, $40 \div 6$, rather than the median.
• G, $6$ — picked a value from the middle of the unsorted list.
• H, $9$ — gave the range, $11 - 2$.

Takeaway: sort, then find the middle. With an even count, average the middle two — and notice the median need not be one of the values in the list.

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Answer: C

Explanation

Go back to totals. The mean of a combined group is the combined total divided by the combined count:

$$\text{class 1 total} = 20 \times 68 = 1360$$ $$\text{class 2 total} = 30 \times 78 = 2340$$

$$\text{overall mean} = \frac{1360 + 2340}{20 + 30} = \frac{3700}{50} = 74$$

Why each wrong option is wrong:

• A, $73$ — averaged the two averages, giving 73. That would only be right if the classes were the same size. The bigger class has the higher average, so the true answer must be pulled above 73 — which is a useful check.
• B, $146$ — added the two averages.
• D, $50$ — gave the total number of students.

Takeaway: averages cannot be averaged unless the groups are equal in size. Convert each to a total, add the totals, divide by the combined count.

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Answer: F

Explanation

Work out both, before and after.

Before, the five values total 70, so the mean is $70 \div 5 = 14$. Sorted, they are $12, 13, 14, 15, 16$, so the median is also 14.

After adding 60, the six values total 130, so the mean is $130 \div 6 = 21.67$. Sorted, they are $12, 13, 14, 15, 16, 60$, so the median is $\frac{14 + 15}{2} = 14.5$.

The mean moved by nearly 8; the median moved by a half.

The reason is structural: the mean uses the size of every value, so one huge number drags it a long way. The median only cares about position, so a huge number counts for no more than a slightly-bigger-than-average one.

Why each wrong option is wrong:

• G, The median rises by much more than the mean does — has it exactly backwards.
• H, Both rise by the same amount — they do not move together; that is the whole point of the question.
• J, Neither the mean nor the median changes — both change, just by very different amounts.

Takeaway: an outlier drags the mean and barely moves the median. That is why house prices and salaries are reported as medians.

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Section 6 — Expressing Numbers in Different Ways

The last part of this category is small but reliable: the same number written as a fraction, a decimal, a percentage, or in scientific notation, and questions about which of several values is largest.

$$\frac{3}{8} = 0.375 = 37.5\% \qquad 4\,500\,000 = 4.5 \times 10^{6}$$

In scientific notation the first number is always at least 1 and less than 10. If you write $45 \times 10^5$ you have not finished.

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Answer: B

Explanation

Move the decimal point until exactly one non-zero digit stands in front of it:

$$4\,500\,000 \longrightarrow 4.5$$

Count how many places it moved: 6. So the number is

$$4.5 \times 10^{6}$$

Check by expanding: $4.5 \times 1\,000\,000 = 4\,500\,000$. Correct.

Why each wrong option is wrong:

• A, $45 \times 10^{5}$ — the right size, but not scientific notation: the leading number must be below 10.
• C, $0.45 \times 10^{7}$ — also the right size, but the leading number must be at least 1.
• D, $4.5 \times 10^{5}$ — counted one place too few, giving 450 000.

Takeaway: scientific notation needs exactly one non-zero digit before the decimal point. Two answers can be the same number and still only one of them is in the right form.

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Answer: H

Explanation

Convert both bounds:

$$\frac{75}{2} = 37.5 \qquad \frac{131}{3} = 43.67$$

The integers strictly between 37.5 and 43.67 are

$$38, \; 39, \; 40, \; 41, \; 42, \; 43$$

That is 6 of them.

The reliable way to count a run of consecutive integers is $\text{last} - \text{first} + 1$:

$$43 - 38 + 1 = 6$$

The $+1$ is the fencepost: a fence from post 38 to post 43 has six posts, not five.

Why each wrong option is wrong:

• F, $4$ and J, $5$ — fencepost errors, counting the gaps between the numbers rather than the numbers themselves.
• G, $7$ — included 37 or 44, which lie outside the range.

Takeaway: convert the bounds to decimals, write down the first and last integer that fit, then use $\text{last} - \text{first} + 1$.

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Answer: A

Explanation

Area first:

$$8 \times 3 = 24 \text{ square metres}$$

Then divide by what one tin covers:

$$\frac{24}{6} = 4 \text{ tins}$$

Here it divides exactly. When it does not, you must round up — three and a bit tins means buying four, because half a tin is not sold.

Why each wrong option is wrong:

• B, $24$ — gave the area and stopped.
• C, $22$ — used the perimeter. Paint covers a surface, so it is an area.
• D, $6$ — repeated the coverage figure.

Takeaway: area ÷ coverage, then round up for anything sold whole.

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Answer: H

Explanation

Convert the depth: $20 \text{ cm} = 0.2 \text{ m}$.

$$4 \times 2.5 \times 0.2 = 2 \text{ cubic metres}$$

Why each wrong option is wrong:

• F, $200$ — used 20 metres as the depth. A slab twenty metres deep is a quarry; the sanity check catches this instantly.
• G, $10$ — gave the area of the slab, forgetting the depth.
• J, $\frac{1}{5}$ — gave the depth on its own.

Takeaway: convert every length to the same unit before multiplying, then ask whether the answer is a sensible size.

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Answer: A

Explanation

Divide, then multiply by 100:

$$\frac{3}{8} = 0.375 \qquad 0.375 \times 100 = 37.5\%$$

Why each wrong option is wrong:

• B, $38$ — read the digits 3 and 8 straight off as "38%".
• C, $\frac{800}{3}$ — divided 8 by 3, giving over 250%. A proper fraction is always less than 100%.
• D, $3$ — gave the numerator.

Takeaway: fraction → decimal → percent, in that order. A fraction below 1 must give a percentage below 100.

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Answer: H

Explanation

Put everything into decimals:

$$\frac{5}{8} = 0.625 \qquad 0.6 = 0.600 \qquad 63\% = 0.630 \qquad \frac{3}{5} = 0.600$$

The largest is $0.63$.

Note how close $0.625$ and $0.63$ are — this is why converting all four is necessary rather than eyeballing them.

Why each wrong option is wrong:

• F, $\dfrac{5}{8}$ — $0.625$, second largest by five thousandths.
• G, $0.6$ and J, $\dfrac{3}{5}$ — both are $0.6$, and they are equal to each other.

Takeaway: convert everything to decimals before comparing, and use the same number of decimal places so the comparison is a straight read.

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Answer: A

Explanation

Two decimal places means keeping $4.64$ and looking at the next digit, which is 7. Seven is 5 or more, so the last kept digit rounds up:

$$4.6472 \longrightarrow 4.65$$

Only the digit immediately after matters. The 2 at the end plays no part.

Why each wrong option is wrong:

• B, $4.64$ — cut the number off instead of rounding.
• C, $4.6$ — rounded to one decimal place.
• D, $4.7$ — rounded the 4 up to 5 and then carried on rounding, which compounds one rounding into another.

Takeaway: look at one digit past the place you are keeping. Round once, not repeatedly.

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Answer: H

Explanation

$$\frac{54}{12} = 4.5 \text{ tins}$$

Four tins would leave part of the wall bare, and half a tin cannot be bought, so round up to 5.

Why each wrong option is wrong:

• F, $4$ — rounded down. Four tins cover only 48 of the 54 square metres.
• G, $6$ — rounded up further than needed.
• J, $12$ — repeated the coverage figure.

Takeaway: for anything sold whole — tins, buses, boxes — divide and then round up, whatever the decimal says.

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Answer: A

Explanation

Two decimal places means hundredths:

$$0.35 = \frac{35}{100}$$

Both parts divide by 5:

$$\frac{35}{100} = \frac{7}{20}$$

Check: $7 \div 20 = 0.35$. ✓

Why each wrong option is wrong:

• B, $\frac{7}{2}$ — used tenths, giving 3.5, ten times too big.
• C, $\frac{1}{35}$ — inverted the fraction.
• D, $\frac{3}{5}$ — read only the 3, giving 0.6.

Takeaway: count the decimal places to fix the power of ten — one place is tenths, two is hundredths — then cancel. Always divide back to check.

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Answer: H

Explanation

Reading from the right: 5 is units, 2 is tens, 8 is hundreds, 4 is thousands.

$$8 \times 100 = 800$$

Why each wrong option is wrong:

• F, $8$ — gave the digit rather than its value.
• G, $80$ and J, $8000$ — counted the places wrongly by one in each direction.

Takeaway: a digit's value is the digit multiplied by its place. Count places from the right: units, tens, hundreds.

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Answer: A

Explanation

Area first:

$$3.5 \times 4 = 14 \text{ square metres}$$

Each tile covers 0.25 m², so

$$\frac{14}{0.25} = 56 \text{ tiles}$$

Dividing by a number below 1 makes the answer bigger, which is the check that catches most errors here — you need more tiles than square metres.

Why each wrong option is wrong:

• B, $14$ — gave the area and stopped.
• C, $28$ and D, $224$ — divided by the wrong tile size.

Takeaway: area ÷ coverage. Dividing by a fraction increases the count, so a smaller answer than the area means something has gone wrong.

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Answer: J

Explanation

Convert everything to decimals:

$$\frac{7}{20} = 0.350 \qquad 0.36 = 0.360 \qquad 34\% = 0.340 \qquad \frac{1}{3} = 0.333$$

The smallest is $\frac{1}{3}$.

Write them all to the same number of decimal places — the four values span only 0.027, so a rough comparison will not separate them.

Why each wrong option is wrong:

• H, $34\%$ — $0.34$, and the closest rival, but still above $0.333$.
• F, $\dfrac{7}{20}$ — $0.35$.
• G, $0.36$ — the largest of the four.

Takeaway: convert to decimals, pad to equal length, then compare digit by digit. $\frac{1}{3} = 0.333\ldots$ is worth knowing on sight.

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Answer: C

Explanation

Unroll the side: it is a rectangle as tall as the pillar and as wide as the circle around it.

$$\text{width} = 2\pi r = 2\pi(0.5) = \pi \qquad \text{height} = 4$$

$$\text{area} = \pi \times 4 = 4\pi \text{ square metres}$$

Why each wrong option is wrong:

• A, $\pi$ — used $\pi r^2 h$, which is a volume. Check the units: an area needs two lengths, not three.
• B, $2 \pi$ — gave the circumference alone.
• D, $8 \pi$ — used 1 as the radius, which is the diameter.

Takeaway: a cylinder's curved surface is $2\pi r h$ — circumference times height. Picture unrolling it, and the formula stops needing to be remembered.

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Answer: F

Explanation

The first significant figure is the 4 — leading zeros never count. So two significant figures means keeping $0.0047$, and the next digit decides whether the 7 rounds up.

That next digit is 3, which is below 5, so the 7 stays:

$$0.004736 \longrightarrow 0.0047$$

Why each wrong option is wrong:

• G, $0.0048$ — rounded up, but the deciding digit is 3, not the 6 further along.
• H, $0.0$ — counted decimal places rather than significant figures.
• J, $0.005$ — gave one significant figure.

Takeaway: significant figures start at the first non-zero digit. Look at the one digit past where you stop, and ignore everything after it.

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Answer: B

Explanation

The fence follows the perimeter:

$$2(12 + 7) = 38 \text{ metres}$$

$$38 \times 9 = 342$$

Why each wrong option is wrong:

• A, $38$ — gave the perimeter and stopped.
• C, $756$ — used the area. A fence goes round an edge; area would be for turfing it.
• D, $171$ — added the two sides once and forgot the garden has two of each.

Takeaway: round an edge is perimeter; covering a surface is area. Decide which before choosing a formula.

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Answer: H

Explanation

$$6\% = \frac{6}{100} = 0.06$$

Moving the decimal point two places to the left is the same operation, and it is quicker: $6.0 \rightarrow 0.06$.

Why each wrong option is wrong:

• F, $0.6$ — moved the point one place, giving 60%.
• G, $6$ — left the number unchanged, so it now means 600%.
• J, $\frac{3}{5000}$ — divided by 100 twice.

Takeaway: percent to decimal divides by 100 — two places left. Check by converting back.

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Answer: A

Explanation

$$\frac{5}{11} = 0.454545\ldots$$

The digits 45 repeat. To three decimal places, look at the fourth digit, which is 5 — that is 5 or more, so the third place rounds up:

$$0.4545\ldots \longrightarrow 0.455$$

Why each wrong option is wrong:

• B, $0.454$ — cut the number off at three places instead of rounding.
• C, $0.045$ — a decimal-place slip; $\frac{5}{11}$ is close to a half, not to a twentieth.
• D, $0.545$ — divided the wrong way round.

Takeaway: a repeating decimal still rounds by the ordinary rule — look at the next digit. And sanity-check the size: $\frac{5}{11}$ is a little under $\frac{1}{2}$.

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