Fractions, Decimals and Percentages Didn’t Have to Be This Hard

A fraction is a whole cut into equal pieces: the bottom number says how many pieces make the whole (a bigger bottom means smaller pieces), and the top counts how many you have. A fraction is also a division (3/4 is 3 ÷ 4) and a point on the number line; a top bigger than the bottom means more than one whole.

Multiplying or dividing the top and bottom by the same number renames a fraction without changing its amount (4/6 = 2/3). That one move simplifies a fraction to lowest terms and lets you compare two fractions by cutting them into the same size of piece; cross-multiplying is a shortcut for the same thing.

You can only add or subtract pieces of the same size, so rename the fractions to a common bottom, then add or subtract the tops and keep the bottom. With mixed numbers, do the wholes and the fractions separately, borrowing a whole when the fraction being taken away is bigger.

"Of" means times. Multiply the tops and the bottoms: the area model shows the bottoms counting the pieces the whole is cut into and the tops counting the pieces you take. Convert mixed numbers to improper fractions first, cancel common factors early, and expect a smaller answer when you multiply by a fraction under 1.

Division asks how many of one thing fit into another. With a common bottom it's whole-number division; flipping the divisor works because the flip of a fraction is how many of it fit into one whole. Dividing by anything less than one makes the answer bigger, and the fractional part of an answer is a fraction of the divisor.

A decimal is a fraction whose bottom is 10, 100 or 1,000, shown by the place of each digit after the point: tenths, hundredths, thousandths. Compare decimals by lining up the points and padding with zeros; turn one back into a fraction by saying it in words and simplifying.

To add or subtract, line up the points so tenths meet tenths; count up from the price to check change. To multiply, ignore the points, multiply the digits, then put back as many decimal places as the two numbers had together, and check with a round-number estimate.

To divide by a decimal, slide the point in both numbers until the divisor is whole. Any fraction becomes a decimal by dividing top by bottom; it ends only if the bottom (in lowest terms) is made of 2s and 5s, and otherwise repeats, because the remainders must come round again.

Round by looking at the one digit after the place you keep (5 or more rounds up). Round once, at the end; rounded parts may not add to the rounded total, so let one part take the leftover. Round up or down when the job demands it, and never report more digits than the inputs deserve.

A percentage is a fraction with 100 on the bottom, so any fraction becomes one by dividing and moving the point two places, and every percentage lines up on the same scale.

Find 10% by moving the point, then build any percentage from tens, halves and ones; swap the numbers (8% of 25 = 25% of 8) whenever that is easier, and multiply when it's a percent of a percent.

Part = percent × whole, so the whole is the part divided by its percentage: undo a discount or a tax by dividing by its multiplier (0.75 after 25% off, 1.0725 with 7.25% tax), never by adding the percentage back.

A percent change is the change divided by the starting value, and every change is a multiplier (× 1.05 up 5%, × 0.85 down 15%): equal rises and falls don't cancel, points differ from percent, and undoing a change means dividing by its multiplier.

When percentages pile up, multiply their multipliers: stacked discounts come to less than their sum, yearly rises compound to more than their sum, equal ups and downs lose a little, and only fixed-dollar amounts make the order matter.

A ratio compares parts with parts; add the parts to find the whole, work out what one part is worth, and every share follows. "2 : 3" means two out of five, never two out of three.

A rate is one quantity divided by another; make sure the "per" is the thing you actually use, flip a rate when the other way up answers your question, put the cost on top when comparing savings, and add rates, never times.

When two amounts grow together, find the amount for one and multiply (cross-multiplying is the same thing in one line); but first check for a fixed part, which doesn't scale, and for amounts that shrink as the other grows.

Scale every length by k and the area is multiplied by k × k and the volume by k × k × k, so a pizza a third wider has 78% more pizza, a rug twice as long each way covers four times the floor, and a box twice as big holds eight times as much.

Before you calculate, or trust a calculation, round to friendly numbers and get a rough answer. Round in opposite directions when you multiply, use compatible numbers for division, and bracket when you can't estimate in one step; if the exact answer isn't close to the estimate, look again.

Every fair split is the whole times your part divided by all the parts. Name the whole, agree what to share it by (people, what each had, square feet, miles, nights, days), and work each share from the total; tax and tip are one multiplier, and settling up only needs each person's difference from the fair share.

When things are different sizes, don't average their percentages, prices or speeds: add up the counts underneath and divide total by total. Mixtures are weighted averages of their parts, a blend's amounts go in the opposite ratio to their distances from the target (the seesaw), and averages are easiest to handle as totals.

Ten habits do almost all the work: name the whole; same-size pieces before you add; "of" means times; dividing asks how many fit; line up the point and count the places; round at the end; ask "percent of what?"; changes multiply; keep parts of parts apart from parts of the whole; and estimate first.