Division and fractions — quotient and remainder, fraction, decimal

The two cards on the left are the dividend (the number being divided, 13) and the divisor (the number you divide by, 4). The four cards on the right show, from the top, the quotient, the remainder, the fraction and the decimal. Changing either number updates all four.

In the quotient card, quot gives the whole-number part of the division; in the remainder card, rem gives what is left over. The fraction card takes a numerator and a denominator and shows the fraction: 13/4 appears as the mixed number 3 1/4, next to the decimal 3.25 and a tape diagram.

A fraction like 13/4, whose top is larger than its bottom, is called an improper fraction. A whole number plus a fraction, like 3 1/4, is a mixed number. The two are the same value.

Why the three forms are the same amount

Suppose 13 is shared equally among 4 people. With 3 given to each person first, 4 × 3 = 12, and 1 is left. That is "3 remainder 1".

If the leftover 1 can be split as well, each person gets another 1/4 of it. One share is then 3 and 1/4 together, 3 1/4. And 1/4 is 0.25, so 3 1/4 is 3.25.

So the fraction and decimal answers are the remainder divided once more by the divisor, added to the quotient: quotient + remainder ÷ divisor = 3 + 1 ÷ 4 = 3.25.

Which form fits depends on what is being shared. 13 apples handed out whole to 4 people means 3 each with 1 left over. A 13 m rope cut into 4 equal pieces gives pieces 3.25 m long.

The remainder is smaller than the divisor

On Board 2 the quotient is entered by hand, on the Guess card. Changing the guess recalculates the remainder as dividend − divisor × guess.

With a guess of 2, the remainder is 5. That is more than the divisor, 4, so everyone could be given 1 more. The "Remainder ÷ divisor" card shows 1 1/4, which is more than 1.

With a guess of 4, the remainder is −3. Giving 4 people 4 each takes 16, and 13 is 3 short.

Only a guess of 3 leaves a remainder that is at least 0 and smaller than 4. The quotient of a division is the one number that puts the remainder in that range, which is why the remainder is always smaller than the divisor.

At that point "Remainder ÷ divisor" is 1/4, less than 1: exactly the fraction part of the mixed number 3 1/4. The card at the top right, guess + remainder ÷ divisor, stays at 3.25 whatever the guess. Handing things out differently does not change the size of one share.

Checking a division

The Check card computes divisor × quotient + remainder: 4 × 3 + 1 = 13, back to the dividend. This is the usual way to check a quotient and a remainder.

It also gives 13 when the guess is 2 or 4, because 4 × 2 + 5 and 4 × 4 + (−3) are both 13. The check on its own does not pin down the quotient. Together with "the remainder is smaller than the divisor", it does.

Back from a decimal to a fraction

3.25 has two decimal places. Multiplied by 100 it is 325, so 3.25 can be written as 325/100. Board 3 reduces that fraction.

Reducing (or simplifying) a fraction means dividing the numerator and the denominator by the same number. The value does not change. A fraction is also a division, numerator ÷ denominator, and dividing both numbers in a division by the same number leaves the answer alone: 325 ÷ 100 and 13 ÷ 4 are both 3.25.

A number that divides both the numerator and the denominator exactly is a common divisor. The largest one is the greatest common divisor (GCD), and dividing by it reduces the fraction in one step. For 325 and 100 the GCD is 25; gcd in the card's formula computes it.

325 ÷ 25 = 13 and 100 ÷ 25 = 4, so 325/100 becomes 13/4. The fraction card at the bottom shows it as the mixed number 3 1/4 — the same fraction as 13 ÷ 4 on Board 1.

The fraction card on Board 1 reduces before it displays, too. When the fraction can be reduced, the unreduced form is shown small above the result.

Other numbers

The dividend and divisor on Board 1 can be stepped by 1 with the − and + buttons.

Stepping the dividend from 12 to 16, the remainder goes 0, 1, 2, 3, and just before it would reach 4 the quotient goes up by 1 and the remainder drops back to 0. With a divisor of 4, the remainder is always one of 0 to 3.

14 ÷ 4 leaves 2, and 2/4 reduces to 1/2, so the fraction is 3 1/2. When the dividend is smaller than the divisor, as in 3 ÷ 4, the quotient is 0 and the remainder is 3.

10 ÷ 3 as a decimal is 3.333… with the 3s going on forever; the board's 3.333333 is cut off. As a fraction it is exactly 3 1/3.

The check and guess + remainder ÷ divisor are the same on every row. Only the row with a guess of 3 has a remainder of at least 0 and below 4.

1.625 has three decimal places, so it becomes 1625/1000. Reduced, it is 13/8 — the same fraction as 13 ÷ 8 on Board 1.

What this covers, and what it does not

A decimal turns back into a fraction this way only when it ends. With 10 ÷ 3 cut off at 3.333 and 3333/1000 entered on Board 3, the GCD is 1, nothing reduces, and the result is not 3 1/3. 3.333 is a different number, slightly smaller than 3 1/3.

The quotient and remainder on the boards are for whole numbers of 0 and above. How to define a remainder for decimals or negative numbers is not covered here.

Read next — Rounding — to the nearest, up and down It rounds numbers to the nearest, up and down, and shows which one applies when a division has to end in a whole number, such as the buses needed or the full boxes.

---

More study examples →