Numerical reasoning, oriented

Most short problems name their operation in the wording: "left" means subtract, "each" with a count means multiply, and "split evenly" means divide. When a quantity changes several times, apply the changes in order to a running value instead of adding every number in sight.

For a list total, adding left to right forces a carry at almost every step. Look first for pairs whose ones digits make ten: in 23 + 41 + 17 + 29, the pairs 23 + 17 = 40 and 41 + 29 = 70 give 110 at once. When a number sits just below a ten, round it and correct: 59 + 26 is 60 + 26 − 1 = 85. When nothing pairs, add all the tens, then all the ones.

Then check. Count the values to be sure none was skipped, the most common slip in list totals. Compare the answer with a rough estimate made from rounded values; an answer far from it signals a mistake. Finally, reread the question. Many wrong choices are right answers to a different question, such as the change instead of the new amount, or the larger share instead of the smaller. Make sure the answer is in the units the question asks for.

Rule: decide the operation, take the fastest route through the arithmetic, and check the result against an estimate and the exact question before choosing.

Ten percent of a number is the number divided by 10, and most other percents build from it: 5% is half of that, 20% is double, and 15% is 10% plus 5%. Several percents are simple fractions: 25% is a quarter and 20% a fifth. A percent of a number also equals that number as a percent of the first, so 4% of 75 is 75% of 4, which is 3.

Apply a change by multiplying. A 15% increase multiplies by 1.15 and a 15% decrease by 0.85, so $36 rising 15% becomes $41.40. Check the direction: a rise must give a larger number and a discount a smaller one. Read whether the question wants the change ($5.40) or the new amount ($41.40), since both are usually among the choices.

Successive changes multiply. A 10% discount followed by a 30% discount leaves 0.9 × 0.7 = 0.63 of the price, a 37% reduction, not 40%. A 30% markup followed by a 20% discount gives 1.3 × 0.8 = 1.04, a 4% gain. An equal percent gain and loss on two items sold at the same price add up to a net loss, because the item sold at a loss cost more.

To reverse a change, divide by its factor. A sale price of $68 after 15% off came from 68 ÷ 0.85 = $80. Adding 15% to the sale price instead gives $78.20, because the discount was taken from the original price, not from the sale price.

Rule: turn every percent change into a multiplier; multiply to apply changes in sequence, and divide to undo them.

Simple interest is a fixed percent of the original amount, added every year, so the balance grows in a straight line. If a deposit doubles in 4 years at simple interest, it earns 25% a year, and reaching three times the deposit, which takes 200% interest, takes 8 years, not 12.

Compound interest applies the rate to the growing balance, so each period's interest is larger than the last. $1,000 at 6% compounded yearly becomes $1,060 after one year and $1,123.60 after two, compared with $1,120 under simple interest. Over n periods the balance is the deposit times (1 + rate) raised to the power n. Savings bonds that compound twice a year apply the same idea every six months, each time to a balance that already includes the interest earned.

Decay is compounding with a factor below one. A value that loses 10% a year keeps 0.9 of itself each year, so after two years it stands at 0.81 of the start, a 19% loss rather than 20%.

When something multiplies by a fixed factor each step, work backward from the end. If a population doubles every year and reaches 8,000 in year 6, it was 4,000 in year 5 and 2,000 in year 4. Each step back divides by the factor.

Rule: simple interest adds a constant, compound interest and decay multiply by a constant, and questions about earlier stages of growth are easier to answer backward.

A ratio such as 3 : 4 divides a quantity into equal parts, so find what one part is worth from whatever you know. A total of $280 in the ratio 3 : 4 is 7 parts of $40, so the shares are $120 and $160. A difference works the same way: if the larger group has 15 more, the gap of 1 part is worth 15, and the total is 7 × 15 = 105. Divide by the sum of the parts for a total and by the difference of the parts for a gap; dividing by one ratio number on its own is the common error.

When only one group changes, anchor on the other. If a ratio of 2 : 3 becomes 2 : 4 after 10 more are added to the second group, the first group is unchanged, so the extra part is worth 10.

Proportions move in one of two directions. Recipes and prices scale directly: if 5 tickets cost $35, 8 cost $56. Fixed jobs scale inversely: if 3 machines take 8 hours, 6 machines take 4, because machines times hours stays constant.

Fractions follow the same logic. If adding 21 liters takes a tank from one fifth full to half full, the 21 liters are 3/10 of the tank, which therefore holds 70 liters. For a problem stated in words, name the unknown, write one equation, and check the answer against every condition.

Rule: find one part, one unit, or one fraction first, then scale, and check whether the scaling should go up or down.

Every travel problem rests on distance = speed × time, rearranged for whichever quantity is missing. Convert first so the units match: 30 minutes is 0.5 hours, and kilometers per hour become meters per second when multiplied by 5/18. A train has crossed a bridge only when its last car leaves it, so the distance covered is the bridge plus the train.

Average speed is total distance over total time, never the average of the speeds. Ride 30 miles at 10 mph (3 hours) and 30 miles back at 30 mph (1 hour), and the average is 60 ÷ 4 = 15 mph, not 20. For a boat, the current adds to its speed downstream and subtracts from it upstream, so the current's speed is half the difference between the two.

Work problems add rates, not times. A printer that finishes a job in 7 hours does 1/7 of it per hour, and one that takes 42 hours does 1/42. Together they do 1/7 + 1/42 = 1/6 per hour, so the job takes 6 hours. A drain or a leak subtracts its rate.

Events that repeat on different cycles first coincide at the least common multiple of the cycles. Buses leaving every 6 and every 10 minutes leave together every 30 minutes.

Rule: write the rate relationship first, match the units, and combine rates by adding them, never by averaging them.

The mean is the total divided by the number of values, so most average problems are really total problems. If four scores average 85, they total 340; if a fifth score brings the average to 86, the five total 430, so the fifth score is 90.

Groups of different sizes combine through their totals, not their averages. A class of 10 averaging 70 and a class of 30 averaging 90 have a combined average of (700 + 2,700) ÷ 40 = 85, not 80, because the larger group pulls the average toward its own.

Mixtures are the same idea. Combining 3 liters of a 35% solution with 4 liters of a 7% solution gives 1.05 + 0.28 = 1.33 liters of the substance in 7 liters, a 19% mixture. When water evaporates or is added, the dissolved amount stays fixed: 2 liters of salt that make up 4% of a solution sit in 50 liters, and if they become 5% of it, the solution is down to 40 liters.

Rule: convert every average and percentage into a total amount, combine the totals, then divide by the new count or volume.