Numerical Reasoning curriculum

Most short word problems describe one operation, and the wording names it. "Left" or "remain" means subtract, "in total" means add, "each" with a count means multiply, and "split evenly" means divide. A shelf of 12 jars after 5 are sold holds 12 − 5 = 7. Name the operation before computing; the classic slip is adding when the story takes something away.

When a quantity changes several times, apply each change in order to a running value. A train car with 30 riders loses 12 at one stop and gains 5 at the next: 30 − 12 = 18, then 18 + 5 = 23. Writing each intermediate value prevents the common error of adding every number, or subtracting every number, whatever its direction.

"How many more" asks for a gap: the target minus the current amount. Counting up is often faster than subtracting. From 68 to 100, go 68 to 70 (2), then 70 to 100 (30), for 32 in all. The trap answer is usually the target or the current amount instead of the gap between them.

Equal groups multiply: 6 boxes of 8 pencils hold 6 × 8 = 48. Fair sharing divides: 48 pencils split among 4 students is 48 ÷ 4 = 12 each. Ask whether the story builds a total from equal groups or breaks a total into equal parts. Sharing should give a smaller number than the total; grouping, a larger one.

Scan a list for pairs whose ones digits add to 10, add those first, and the rest is easy. In 17 + 26 + 13 + 24, pair 17 + 13 = 30 and 26 + 24 = 50, for a total of 80. Adding left to right forces a carry at almost every step; pairing removes most of them, and with them most of the slips.

When a value sits just below a ten, round it up, add, and take back what you added. For 48 + 37, treat 48 as 50: 50 + 37 = 87, then take back 2 for 85. Subtraction works the same way: 92 − 29 is 92 − 30 + 1 = 63. Track the correction and its direction, or the answer is off by exactly that amount.

Split each number into tens and ones and add each group separately. For 34 + 25 + 18, the tens give 30 + 20 + 10 = 60 and the ones give 4 + 5 + 8 = 17, so the total is 60 + 17 = 77. This route always works, even when no pair rounds neatly, and it leaves the only carry for the final step.

Before adding, count how many values the list has, and check that count against the values you actually used. Skipping one value is the most common error in list totals, and it produces an answer that still looks reasonable. Ticking off each number as you add it catches an omission that rechecking the arithmetic never will.

Round each value to the nearest ten and add those first: 38 + 41 + 29 is about 40 + 40 + 30 = 110. The exact total, 108, should land near the estimate, so a result of 128 or 88 signals a slip. An estimate also rules out answer choices that cannot be right before any exact work is done.

Turn a large number of minutes into hours and minutes first: 500 minutes is 8 hours 20 minutes, since 8 × 60 = 480. Then add to the clock reading, carrying every 60 minutes into an hour and wrapping past 12 or 24. From 07:15, adding 8 hours 20 minutes gives 15:35. Treating minutes as decimal hours gives the wrong time.

Ten percent of a number is the number divided by 10: 10% of 370 is 37. From there, 5% is half of that (18.5), 20% is double (74), and 30% is triple (111). Building percents from 10% avoids long multiplication, and each step is a simple halving, doubling, or tripling that is easy to check.

Several percents are simple fractions: 50% is one half, 25% one quarter, 75% three quarters, 20% one fifth, and 12.5% one eighth. So 25% of 64 is 64 ÷ 4 = 16, and 20% of 45 is 45 ÷ 5 = 9. Recognizing the fraction turns a percent problem into one division.

Any percent of a number equals that number percent of the first, because both are the two numbers multiplied together and divided by 100. So 8% of 25 is the same as 25% of 8, which is 2. When one order is awkward, the other is often a benchmark, and the swap is a quick check on an answer found the long way.

To find p% of an amount, divide p by 100 and multiply: 35% of 260 is 0.35 × 260 = 91. This works when no benchmark fits. Place the decimal carefully, since 0.35 and 0.035 differ by a factor of ten, and estimate first: 35% is a little more than a third, so expect a little more than 260 ÷ 3, about 87.

Percent questions ask for two different things. "How much do you save on a $48 jacket at 25% off?" wants the change, $12. "What do you pay?" wants the new amount, $36. Both numbers usually appear among the choices, so reread the final question before choosing.

A p% increase multiplies the original by (1 + p/100). An $84 item rising 25% becomes 84 × 1.25 = $105. This equals finding the rise, $21, and adding it, but the multiplier is faster when changes repeat. Typical wrong choices are the rise alone, $21, or the original plus the rate read as dollars, $109.

A p% decrease multiplies the original by (1 − p/100). A $250 price cut by 20% becomes 250 × 0.8 = $200. Thinking "I keep 80%" is often quicker than computing the cut and subtracting it. Watch the multiplier for small rates: a 5% cut multiplies by 0.95, not 0.5.

After any percent change, check the direction and the size. A price that rose should be higher, and a discounted price lower. A 10% change moves a number by a tenth of itself, not by 10: a $64 item raised 10% should cost $70.40, so an answer of $74, which adds $10 instead of 10%, fails the check.

Two percent changes in a row multiply rather than add. A 20% discount followed by 10% more off leaves 0.8 × 0.9 = 0.72 of the price, a 28% total discount, not 30%, because the second cut applies to an already reduced amount. Multiply the factors, then convert back to a percent if the question asks for one.

A markup and a later discount do not cancel by subtraction. Mark a $200 item up 30% to $260, then take 10% off: 260 × 0.9 = $234, a 17% profit on cost, not 20%. The net factor is (1 + markup) × (1 − discount), here 1.3 × 0.9 = 1.17.

Selling two items at the same price, one at a 20% profit and one at a 20% loss, does not break even. The item sold at a loss cost more, so its loss is larger in dollars. At $96 each, the first cost $80 and the second $120: the seller takes in $192 on $200 of cost and loses $8.

A price of $64 after a 20% discount came from 64 ÷ 0.8 = $80, not from adding 20% of $64 to get $76.80. The percent was taken from the original, so undo it by dividing by the same factor. An increase reverses the same way: $66 after a 10% rise came from 66 ÷ 1.1 = $60.

To recover a price before two discounts, divide by the product of both factors. After 25% and then 10% off, an item costs 0.75 × 0.9 = 0.675 of its original, so a final price of $243 means the original was 243 ÷ 0.675 = $360. Adding the percents to 35% and dividing by 0.65 gives about $374, which is wrong.

For deals such as "buy 3, get 1 free," divide what you pay by how many items you get. Four bottles at $6 under that offer cost 3 × $6 = $18, an effective $4.50 each. Wrong routes include dividing by the number paid for rather than the number received, or taking one item's price off a single unit.

Simple interest adds the same amount each year: a fixed percent of the original deposit. A deposit that doubles in 5 years earns 100% of itself in 5 years, or 20% a year. To reach four times the deposit it must earn 300%, which takes 15 years. Time scales with the total interest earned, not with the final multiple.

Compound interest applies the rate to the growing balance. $640 at 5% becomes 640 × 1.05 = $672 after one year and 672 × 1.05 = $705.60 after two, so the interest is $65.60, not the $64 that simple interest would pay. Over n years the balance is the deposit times (1 + rate) to the nth power.

A value that loses a fixed percent each year shrinks by a factor each year, so the dollar losses get smaller over time. A $20,000 car losing 15% a year is worth 20,000 × 0.85 = $17,000 after one year and 17,000 × 0.85 = $14,450 after two. Subtracting 15% of the original twice, to $14,000, overstates the loss.

When something grows by a fixed factor each step, work back from the known end. If a culture doubles every hour and fills a dish at hour 10, the dish was half full at hour 9 and a quarter full at hour 8, not at hour 5. Each step back divides by the growth factor, so one eighth full sits three doublings before full.

A ratio such as 2 : 5 splits a quantity into equal parts, 2 of one kind and 5 of the other. Every ratio problem gets easier once you know what one part is worth. Divide the quantity you know, whether a total or a difference, by the number of parts it covers, then multiply by the parts you need.

To split $360 in the ratio 4 : 5, add the parts to get 9, so one part is 360 ÷ 9 = $40. The shares are 4 × 40 = $160 and 5 × 40 = $200, which add back to $360. The common error is dividing the total by one of the ratio numbers instead of by their sum.

If boys and girls are in the ratio 2 : 7 and there are 35 more girls, the difference is 7 − 2 = 5 parts, so one part is 35 ÷ 5 = 7. That gives 14 boys and 49 girls. Dividing the difference by either ratio number on its own, rather than by the difference in parts, is the usual mistake.

Once a difference has given you one part, the total is the sum of the ratio numbers times that part. With a ratio of 2 : 7 and one part worth 7, the total is 9 × 7 = 63. Questions often give a difference and ask for a total, so carry the part value through rather than stopping at one group.

Divide both numbers of a ratio by their greatest common factor: 12 : 18 becomes 2 : 3, and 15 : 25 becomes 3 : 5. Simpler ratios make the part sizes easier to see and the arithmetic smaller. Simplifying does not change the comparison, so either form gives the same answer if the arithmetic is right.

When items are added to only one group, the other group stays the same, so anchor on it. Cats and dogs at 2 : 3 become 2 : 5 after 12 more dogs arrive. The cats did not change, so the dogs went from 3 parts to 5 parts of the same size: 2 parts equal 12, one part is 6, and there are 12 cats.

When two quantities rise together at a fixed rate, find the amount for one unit, then scale. If 3 notebooks cost $7.50, one costs $2.50, so 8 cost $20. You can also scale the ratio directly: 8 is 8/3 of 3, so the cost is 8/3 of $7.50. Check that more items cost more.

When the total work is fixed, more workers finish sooner, because workers times time stays constant. If 4 cooks prepare a banquet in 9 hours, the job is 36 cook-hours, so 6 cooks need 36 ÷ 6 = 6 hours. Scaling up as if it were a direct proportion, to 13.5 hours, gets the direction wrong.

Distance, speed, and time are linked by one formula: distance = speed x time. Rearranged, speed = distance / time and time = distance / speed. Write down which two quantities you know and which one you need, and keep the units matched, such as miles with miles per hour and hours.

Average speed is total distance divided by total time: 225 miles in 4.5 hours is 225 ÷ 4.5 = 50 mph. Rewriting the time can help: 4.5 hours is 9 half-hours, so each half-hour covers 25 miles. Multiplying distance by time instead gives an absurdly large speed, which a quick reality check catches.

Distance is speed multiplied by time: a cyclist riding 14 miles per hour for 2.5 hours covers 14 × 2.5 = 35 miles. Put the time in the same unit as the speed first: 2 hours 30 minutes is 2.5 hours, not 2.3. Riding faster or longer should always give a longer distance.

Time is distance divided by speed: 210 miles at 60 mph takes 210 ÷ 60 = 3.5 hours, which is 3 hours 30 minutes. Turn the decimal part into minutes by multiplying it by 60. A higher speed over the same distance must take less time; if your answer grows when the speed grows, the formula was inverted.

Rates combine only in matching units. To turn kilometers per hour into meters per second, multiply by 1,000 and divide by 3,600, which simplifies to multiplying by 5/18: 108 km/h is 30 m/s. Likewise, 45 minutes is 0.75 hours. Most errors in rate problems come from mixing minutes with hours or meters with kilometers.

A train has cleared a bridge only when its last car leaves the far end, so the distance covered is the bridge length plus the train length. A 120 m train at 15 m/s crossing a 180 m bridge travels 300 m, which takes 300 ÷ 15 = 20 seconds. Using only the bridge length gives 12 seconds, which is too short.

Average speed is total distance over total time, not the average of the two speeds. Drive 84 miles out at 42 mph (2 hours) and back at 84 mph (1 hour): 168 miles in 3 hours is 56 mph, not 63. The slower leg takes longer, so it pulls the average toward the lower speed.

Downstream, the current adds to a boat's speed; upstream, it subtracts. So the current's speed is half the difference of the two: a boat making 15 km/h downstream and 9 km/h upstream rides a current of (15 − 9) ÷ 2 = 3 km/h, and its own speed is (15 + 9) ÷ 2 = 12 km/h. Turn distances and times into speeds first.

When several workers or pipes act together, add their rates, not their times. A pump that fills a pool in 14 hours does 1/14 of the job per hour; one that takes 35 hours does 1/35. Together they do 1/14 + 1/35 = 1/10 per hour, so the pool fills in 10 hours. A drain works against them, so subtract its rate.

Events repeating every a and b units next coincide after the least common multiple of a and b. Lights flashing every 8 and 12 seconds flash together every 24 seconds, the smallest number both divide evenly. Multiplying the two, to 96, gives a time when they do coincide, but not the first.

The mean is the sum of the values divided by how many there are: scores of 81, 74, and 90 total 245, so the average is 245 ÷ 3, about 81.7. Count the values carefully, since dividing by the wrong number is a common slip, and check that the mean falls between the smallest and largest value.

Averages combine through totals. If 20 workers average 30 years of age and the 8 managers among them average 42, the whole group totals 20 × 30 = 600 years and the managers 8 × 42 = 336. The other 12 workers total 264 years, an average of 22. Averaging the averages directly fails when groups differ in size.

Mixing 4 liters of a 6% solution with 1 liter of a 46% solution gives 0.24 + 0.46 = 0.7 liters of the dissolved substance in 5 liters, a 14% mixture. The plain average of 6% and 46%, 26%, ignores that there is four times as much of the weaker solution. Add the amounts of substance, then divide by the total volume.

When a mixture gains or loses only one ingredient, the other ingredient's amount stays fixed. A 50-liter solution that is 6% salt holds 3 liters of salt. If water evaporates until it is 10% salt, those 3 liters are now 10% of the total, so the total is 30 liters and 20 liters of water evaporated.

To find three fifths of 40, divide by the denominator and multiply by the numerator: 40 ÷ 5 = 8, and 8 × 3 = 24. "Half of" means divide by 2 and "a third of" means divide by 3. Dividing first keeps the numbers small; multiplying first gives the same answer through larger intermediate values.

Fractions can be added or subtracted only over a shared denominator. To compute 3/4 − 1/6, use 12: 3/4 is 9/12 and 1/6 is 2/12, so the difference is 7/12. Subtracting the numerators and the denominators separately gives 2 over -2, which means nothing; the common denominator is the whole technique.

If a tank goes from one third full to five sixths full when 30 liters are added, those 30 liters are 5/6 − 1/3 = 5/6 − 2/6 = 3/6, or one half, of the tank. So the full tank holds 30 ÷ (1/2) = 60 liters. Name the fraction that the known amount represents, then divide the amount by that fraction.

When a problem relates quantities in words, let a letter stand for one of them and translate each statement. "A father is 4 times as old as his daughter; in 10 years he will be 2.5 times as old" becomes 4d + 10 = 2.5(d + 10), so 1.5d = 15 and d = 10: the daughter is 10 and the father 40. Check both statements.

Consecutive integers are n, n + 1, n + 2; consecutive even or odd integers are n, n + 2, n + 4. Three consecutive odd integers summing to 81 give 3n + 6 = 81, so n = 25 and the numbers are 25, 27, and 29. For an odd count of consecutive numbers, the middle one is the sum divided by the count: 81 ÷ 3 = 27.

Many wrong choices are right answers to a different question: the increase instead of the new price, the larger share instead of the smaller, the interest instead of the final balance. Before choosing, reread the last sentence of the problem and check that your number answers it, in the units it asks for.

A rough estimate often removes most choices. Since 19% of 412 is close to 20% of 400, which is 80, choices of 41, 112, and 778 can be dropped at once, leaving 78. Estimating first also guards against decimal slips, which tend to produce answers off by a factor of ten.

Substitute your answer into every condition of the problem. If you found 15 cats and 27 dogs for a 5 : 9 ratio with 12 more dogs than cats, check: 15 : 27 simplifies to 5 : 9, and 27 − 15 = 12. Both conditions hold. When one fails, the error is usually in the step that used that condition.

Check that the answer's unit matches the question: hours rather than minutes, meters rather than kilometers, dollars rather than percent. A time of 2.5 hours is 150 minutes, and a choice of 2.5 minutes is a trap built from the right number in the wrong unit. Convert deliberately, at the end.

A few errors recur: averaging two speeds instead of dividing total distance by total time, adding successive percents instead of multiplying their factors, dividing by one ratio number instead of the sum of the parts, and reporting a change when the question wants the new amount. Knowing them makes the trap choice easy to spot.