Arithmetic Reasoning: mixed review

$x-8$: start with the number and take 8 away. Writing $8-x$ reverses the order and is a classic trap.

$2x+7=31$, so $2x=24$ and $x=12$. 'Is' becomes the equals sign; build 'twice a number' first, then add 7.

When the words say to combine first. 'Twice the sum of a number and 7' is $2(x+7)$, not $2x+7$.

Ask what the story does to the quantities: combining or increasing means add; removing, comparing or finding a missing part means subtract; equal groups or rate times time means multiply; equal sharing or finding a rate means divide.

1.25 hours, not 1.15. Minutes are sixtieths of an hour, so $15÷60=0.25$ hour.

What is left is 60 percent of the original: $0.60x=270$, so $x=450$. Adding 40 percent of 270 (giving 378) is the trap.

Note what is asked and its unit; name the unknown; keep only the facts linked to it; translate into an equation, converting units first; estimate the size and direction; calculate and check against the question and choices.

Intermediate subtotals often appear among the choices. Make sure your number is the final quantity asked for, such as the change from 100 dollars rather than the 54 dollar cost.

Grouping symbols, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

12. Multiplication and division share one level, so go left to right: $18÷3=6$, then $6\times 2=12$. Multiplying first (giving 3) is the PEMDAS trap.

Work from the innermost group outward: $8-5=3$, then $6+3\times 3=15$, then $4\times 15=60$. A number written next to a bracket means multiply.

$347÷8=43$ remainder 3: 43 full vans, 44 vans needed so nobody is left behind, and 3 trainees left over. The question decides which number to report.

3: the digit sum is a multiple of 3. 9: the digit sum is a multiple of 9. 4: the last two digits form a multiple of 4. 6: the number passes both the 2 test and the 3 test.

By 5: times 10, then halve ($68\times 5=340$). By 25: times 100, then divide by 4 ($64\times 25=1600$). By 99: times 100, minus the number ($37\times 99=3663$).

Place the sum of its digits between the digits, carrying 1 into the left digit when the sum is 10 or more: $43\times 11=473$ and $78\times 11=858$.

Multiply the quotient by the divisor and add the remainder; you should get the dividend back. For $380÷32=11$ R 28: $11\times 32+28=380$.

The denominator (bottom) is how many equal parts make the whole; the numerator (top) is how many parts are counted. A fraction is also a division: $\frac{3}{4}=3÷4=0.75$.

Divide the top and bottom by their greatest common factor: $\frac{12}{30}=\frac{12÷6}{30÷6}=\frac{2}{5}$.

Cross-multiply: $4\times 9=36$ and $5\times 7=35$. The larger product goes with $\frac{4}{7}$, so $\frac{4}{7}$ is larger.

Multiply the whole number by the denominator, add the numerator, and keep the denominator: $4\frac{3}{8}=\frac{4\times 8+3}{8}=\frac{35}{8}$.

Denominators are never added. Rename both with a common denominator, then add numerators: $\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$.

Multiply numerators together and denominators together; no common denominator is needed. Cancel shared factors first: $\frac{4}{9}\times \frac{15}{16}=\frac{5}{12}$.

Keep the first fraction, change division to multiplication, and flip only the divisor: $\frac{5}{6}÷\frac{2}{3}=\frac{5}{6}\times \frac{3}{2}=\frac{5}{4}=1\frac{1}{4}$.

No. Convert to improper fractions first: $2\frac{1}{2}\times 3\frac{1}{3}=\frac{5}{2}\times \frac{10}{3}=\frac{25}{3}=8\frac{1}{3}$, not $6\frac{1}{6}$.

Tenths, hundredths and thousandths; each place is worth one tenth of the place to its left. In 36.582, the 8 stands for 8 hundredths, or 0.08.

0.5. Pad with zeros to the same length (0.50 versus 0.45) before comparing; a decimal with more digits is not automatically bigger.

Check the digit just right of the target place: 5 or more raises the target digit by one, 4 or less leaves it. 7.364 becomes 7.4 to the tenth and 7.36 to the hundredth; 2.995 becomes 3.00.

Multiply as whole numbers, then give the product as many decimal places as both factors have in total: $32\times 45=1440$, so $3.2\times 0.45=1.44$, and $0.3\times 0.2=0.06$.

Move the divisor's point right until it is a whole number and move the dividend's point the same number of places: $86.4÷4=21.6$. Shifting only one of them changes the answer.

The decimal point shifts one place per zero, right when multiplying and left when dividing: $3.07\times 1000=3070$ and $52.6÷100=0.526$.

$\frac{1}{8}=0.125$, $\frac{1}{5}=0.2$, $\frac{3}{4}=0.75$, $\frac{7}{8}=0.875$. Any fraction converts by dividing the numerator by the denominator.

Put the digits over the place value of the last digit, then simplify: $0.36=\frac{36}{100}=\frac{9}{25}$.

Part = percent times whole; percent = part divided by whole; whole = part divided by percent, with the percent written as a decimal. The whole usually comes right after 'of'.

Percent to decimal: move the point two places left (7 percent is 0.07, 125 percent is 1.25, 0.6 percent is 0.006). Decimal to percent: move it two places right.

$\text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times 100$. Always divide by the original value: 50 rising to 65 is $15÷50=0.30$, a 30 percent increase.

Divide by the multiplier: $182÷1.30=140$ thousand. Taking 30 percent off 182 (about 127.4) is wrong because the increase was figured on the original.

Discount: multiply by 1 minus the rate (15 percent off means times 0.85). Tax or markup: multiply by 1 plus the rate (6 percent tax means times 1.06).

495, not 500. Multiply the factors in order: $500\times 1.10\times 0.90=495$. The second change is taken from the new, larger amount.

Discount first, then tax the sale price: $120\times 0.80=96$ and $96\times 1.07=102.72$, so 102.72 dollars.

Get 10 percent by moving the point one place left; 5 percent is half of that and 15 percent is their sum; 1 percent moves the point two places. Swapping helps too: 8 percent of 50 equals 50 percent of 8, which is 4.

Part-to-part compares one group with another (8 sedans to 12 trucks is 2:3). Part-to-whole compares a group with the total, which is the sum of the parts (8 to 20 is 2:5).

Convert to the same unit first, then divide by the greatest common factor: 24 inches to 8 inches is 24:8, which is 3:1.

The cross products are equal: $a\times d=b\times c$. For $\frac{5}{12}=\frac{x}{36}$, $12x=180$, so $x=15$. Multiply diagonally, never straight across.

Add the terms to get 7 parts, find one part ($84÷7=12$), then multiply: $2\times 12=24$ and $5\times 12=60$ tools.

The difference is $7-3=4$ parts, so one part is $28÷4=7$. Rifles: 49, pistols: 21, total: 70. Ratio terms are not counts until you find one part.

$k=\frac{\text{new length}}{\text{original length}}$; $k>1$ enlarges and $0<k<1$ shrinks. Areas change by $k^2$: with $k=2.5$, the area becomes 6.25 times as large.

When the quantities do not rise together at a constant ratio, such as workers and time to finish a job: more workers means less time. Cost and quantity at a fixed price are direct.

60 is 5 times 12, so scale the numerator by 5 as well: $x=7\times 5=35$. Look for a whole-number scale factor before cross-multiplying.

Add the values and divide by how many there are: 12, 15, 9, 20 and 14 total 70, and $70÷5=14$. The mean always lies between the smallest and largest values.

$\text{average}\times \text{count}=\text{total}$. Six deliveries averaging 45 minutes take $6\times 45=270$ minutes in all.

Needed total: $4\times 85=340$. Known sum: 251. Fourth score: $340-251=89$.

$\frac{n_1{x}_1+n_2{x}_2}{n_1+n_2}$, where n is each group's size and x its mean. 15 averaging 60 and 25 averaging 68 give $2600÷40=65$. Never just average the averages.

Multiply each value by its weight and add; weights totaling 100 percent need no division: $0.30\times 95+0.70\times 75=81$. If weights are counts or points, divide by their sum.

48 mph: total distance over total time, $240÷(3+2)=48$. Averaging the speeds to get 50 is wrong because more time was spent at the slower speed.

Only when equal time is spent at each speed. When equal distances are covered, the true average is below the plain average because the slower leg takes longer.

Pure amount = concentration times volume. Add the pure amounts and divide by total volume: 6 L at 10 percent (0.6 L) plus 4 L at 35 percent (1.4 L) is 2 L in 10 L, or 20 percent.

The amount of the first quantity for exactly one unit of the second; divide the first by the second. 450 gallons in 15 minutes is $450÷15=30$ gallons per minute.

$d=r\times t$, $r=\frac{d}{t}$, $t=\frac{d}{r}$, where d is distance (miles), r is rate (miles per hour) and t is time (hours). Keep the units matched.

20 miles. Convert minutes to hours first (30 minutes = 0.5 hour). Multiplying 40 by 30 gives 1,200, which is 60 times too large.

Compare the price per unit, not the sticker price. 6 bottles for 4.20 dollars is 0.70 dollar each; 10 for 6.50 dollars is 0.65 dollar each, so the bigger pack is cheaper per bottle.

Add rates of $\frac{1}{h}$ job per hour: $\frac{1}{4}+\frac{1}{12}=\frac{1}{3}$, so 3 hours. A team is always faster than its fastest member.

Use the net rate: $20-4=16$ liters per minute, so $400÷16=25$ minutes. Dividing by the inflow alone is the trap.

Add when the objects move toward each other or apart in opposite directions; subtract when one follows the other in the same direction. Then time = gap divided by that combined speed.

Let units cancel like factors: miles per hour times hours leaves miles; miles divided by miles per hour leaves hours. A result in miles times hours, or hours per mile for a speed, means you mixed up multiply and divide.

$d=r\times t$: distance equals rate times time. Know any two to find the third. It assumes a steady speed, or uses the average speed over the trip.

Convert time to hours first: $72\times \frac{25}{60}=30$ km. Multiplying 72 by 25 directly gives an impossible 1,800.

Multiply the decimal part of the hour by 60: $0.5\times 60=30$, so 5 hours 30 minutes.

The sum of their speeds when approaching or moving apart, the difference in a same-direction chase. Time = gap divided by relative speed: a 57 mph car closes an 18-mile gap on a 48 mph bus in $18÷9=2$ hours.

Turn it into a lead distance. A 45 mph truck leaving 40 minutes early leads by $45\times \frac{40}{60}=30$ miles; a 60 mph car gains 15 mph, so it catches up in 2 hours.

$\frac{2ab}{a+b}$. At 30 mph and 60 mph: $\frac{2\times 30\times 60}{90}=40$ mph, not 45. The slower leg takes longer, so the average leans toward the slower speed.

Add the stop to total time but add no distance. 104 miles in 2 hours, a 30-minute stop, then 90 miles in 1.5 hours gives $194÷4=48.5$ mph.

First remove A's early distance: $260-40=220$ miles. They close at 110 mph, so they meet 2 hours after B starts, which is 3 hours after A starts.

$\frac{1}{h}$ of the job per hour, with the whole job counted as 1. A mechanic who rebuilds an engine in 5 hours does $\frac{1}{5}$ of it each hour.

$W=R\times t$: work done (fraction of the job) equals rate (jobs per hour) times time (hours). At $\frac{1}{9}$ per hour, 4 hours completes $\frac{4}{9}$, leaving $\frac{5}{9}$.

Add their rates, then invert the sum. For 6 and 10 hours: $\frac{1}{6}+\frac{1}{10}=\frac{4}{15}$, so the time is $\frac{15}{4}$ hours, or 3 hours 45 minutes.

Subtract the known rate from the team rate: $\frac{1}{6}-\frac{1}{10}=\frac{1}{15}$, so B alone needs 15 hours.

The drain's rate is subtracted: $\frac{1}{6}-\frac{1}{15}=\frac{1}{10}$ tank per hour, so the tank fills in 10 hours.

Done: $\frac{5}{15}=\frac{1}{3}$, so $\frac{2}{3}$ remains. Team rate: $\frac{1}{15}+\frac{1}{10}=\frac{1}{6}$. Rest: $\frac{2}{3}÷\frac{1}{6}=4$ hours. Total: 9 hours.

Let the job be the LCM of the solo times. For 6 and 9 hours, call it 18 units: the rates are 3 and 2 units per hour, 5 together, so $18÷5=3.6$ hours.

Never add or average times. Add rates: $\frac{1}{3}+\frac{1}{6}=\frac{1}{2}$, so 2 hours. The team time must fall between half the faster worker's time and that worker's time.

$I=P\times r\times t$: I is interest in dollars, P the principal in dollars, r the yearly rate as a decimal, t the time in years. 3,000 dollars at 4 percent for 5 years earns 600 dollars.

Divide months by 12 to get years: 18 months is 1.5 years. 2,400 dollars at 5 percent for 18 months: $2400\times 0.05\times 1.5=180$ dollars.

$r=\frac{I}{P\times t}$. A 2,500 dollar deposit earning 250 dollars in 2 years: $250÷5000=0.05$, or 5 percent per year.

Only hours beyond 40 earn 1.5 times the regular rate. At 24 dollars per hour for 48 hours: $40\times 24+8\times 36=1248$ dollars.

An amount added to the seller's cost to set the price; it is a percent of the cost. A 40 percent markup on a 45 dollar cost adds 18 dollars, so the price is $1.40\times 45=63$ dollars.

Profit = revenue minus cost; a negative result is a loss. 40 shirts bought at 6 dollars and sold at 9 dollars: $360-240=120$ dollars, which is 50 percent of cost.

No, the base changes: 100 dollars drops to 80, then $80\times 1.20=96$. Likewise 25 percent off followed by 8 percent tax is not simply 17 percent off.

Commission is pay equal to a percent of sales. Total pay = base + rate times sales: a 600 dollar base plus 3 percent of 20,000 dollars is $600+600=1200$ dollars.

Pin down exactly what is asked; list the numbers that matter; convert to common units; write the operations before computing; solve in labeled steps; check that the answer is reasonable and matches the target.

180. The 360 left after Monday is only an intermediate value; Tuesday ships 180 of them. Label each result so you do not stop on a stepping stone.

$\frac{5}{8}\times 720=450$, then $0.80\times 450=360$. Apply each percent to the group it describes, not the original total.

$5\times (8+2)=50$ dollars. Writing $5\times 8+2=42$ charges shipping only once; decide whether each charge is per item or per order.

Convert to one unit before combining: 2 yards is 6 feet, so the total is 6 + 4 = 10 feet. Quantities can be added only when their units match.

Undo the change by dividing by the multiplier: $23÷1.15=20$ dollars. Subtracting 15 percent of 23 is wrong because the raise was based on the old wage.

Cross out choices that fail your estimate or have the wrong unit or size, watch for intermediate values planted as choices, and in reverse problems test a choice by running it forward.

$1746÷24=72$ remainder 18, so 72 full boxes. The 18 leftover units do not fill another box, so do not round up.