Seahorse Series · Quiz 8 · What a Fraction Is

The Guide

Before you begin

A fraction is what a number looks like between the whole numbers: a part of a whole, written as one number over another. If you did Quiz 5, you saw three single dollars left on the counter that nobody could take without breaking them into coins; this quiz breaks them. And the factors from Quiz 7 turn out to be the tool that tames fractions. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

The two numbers do two different jobs

In 34, the bottom number, the denominator, names the size of the piece: it says how many equal pieces make one whole. The top number, the numerator, counts how many of those pieces you have. Bottom names, top counts. Nearly every fraction error in the world comes from forgetting that these two numbers are not the same kind of thing.

Drawn to scale. The word equal matters: four pieces that are not equal are not fourths.

A fraction is also an address

Pizzas and pies are how fractions are usually introduced, but that picture can only take you so far. The more powerful picture, and the one the GED leans on hardest, is the number line from Quiz 2: 34 is not only three slices, it is a place, three quarter-steps along the road from 0 to 1. Every fraction has a house on the same street as the whole numbers.

The same road as always, with the quarter-steps now marked. Notice that 7/4 and 1¾ are one address with two names.

One amount, many names

Cut every piece of a half into two, and you have two pieces out of four: the pieces got smaller, but you hold the same amount. That is equivalence, and it is why multiplying the top and bottom by the same number never changes a fraction's value: you are only re-cutting, never adding or taking away.

Drawn to scale. The teal line is the argument: three names, one address.

Simplifying: Quiz 7, put to work

To simplify is to run the re-cutting backward, gluing small pieces back into larger ones. Divide the top and the bottom by a common factor, and keep going until the only factor they share is 1: that is simplest form. For 1824, both divide by 6, giving 34 — and 3 and 4 share nothing further, so the work is done. This is where the divisibility tests from Quiz 7 (the quick checks for whether a number divides by 2, 3, 5, and so on) earn their keep.

Why "reducing" is another misleading word: like borrowing in Quiz 3, it suggests something is lost. Nothing is. The amount is untouched; only the names of the pieces change, from many small ones to fewer large ones. Simplest form is not a smaller fraction, it is the same fraction wearing its plainest clothes.

The fourth fate, at last

Quiz 5 promised this. When 87 dollars is shared by 4 people, long division answers 21 R 3, and we noted that the leftover could itself be cut into equal shares. Now it can: those 3 singles split four ways give each person 34 of a dollar, so the true answer is 21¾. Every improper fraction (a fraction whose top is bigger than or equal to its bottom, like 74) is a division waiting to be done, and every remainder is the numerator of what's left: 74 means 7 ÷ 4, which is 1 R 3, which is 1¾, a mixed number (a whole number and a fraction together).

Worked Examples

Three to study before you start

Example 1 · Building an equivalent name

Write 34 with a denominator of 12. Ask what turns 4 into 12: multiplying by 3. Then the top must take the same treatment, because re-cutting applies to every piece: 3 × 3 = 9. So 34 = 912. Twelve smaller pieces, nine held, same address on the line.

Example 2 · Simplifying with the tests

Simplify 1824. Both are even, so divide by 2: 912. Both digit sums divide by 3, so divide by 3: 34. Now 3 and 4 share only 1, so the work is finished. Taking it in two easy steps is perfectly legal; spotting the 6 at once just gets there faster.

Example 3 · Improper to mixed, by division

Write 114 as a mixed number. The bar means divide: 11 ÷ 4 = 2 R 3. The answer to the division (the quotient), 2, is the whole part; the remainder, 3, goes on top of the same denominator: 2¾. Going back is the same trip reversed: 2 wholes are 8 fourths, plus 3 more makes 114.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Write your answers on the lines. Questions 4 and 8 are multiple choice: choose the one best answer.

A wall is divided into 8 equal panels, and 3 of them have been painted. What fraction of the wall is painted?
In the fraction 58, which number tells you the size of each piece, and which tells you how many you have?
The gold point marks a number on the line.

Write the fraction it marks.
Which bar shows exactly one half shaded?
- A) Bar A
- B) Bar B
- C) Bar C
- D) None of them
Fill in the missing number: 35 = ?20
Simplify: 1540

▶ VideoA video rest stop, for anyone who has kept going this far.

Take two minutes with these seahorses.

Simplify: 2436
Write 74 as a mixed number.
- A) 1¾
- B) 3¼
- C) 7¼
- D) 4/7
Back to Quiz 5's counter: 87 dollars shared equally by 4 people came to 21 R 3. Now cut the leftover as well. Exactly how much does each person get?
Diego's shift is 8 hours long, and he has worked 6 of them. What fraction of the shift has he worked? Give your answer in simplest form.

Answer Key

Check your work

Open the key: after you've finished all ten

1 3/8

Eight equal panels make the whole, so 8 names the size of the piece and sits below; three are painted, so 3 counts and sits above. If you wrote 3/5, you counted the unpainted panels as the whole. The denominator is always the whole thing, never the part you didn't take.

2 The 8 tells the size of each piece; the 5 tells how many.

Bottom names, top counts. Say it about a real object until it's automatic: eighths are the size, and you have five of them. Every later fraction skill leans on this one sentence.

3 5/8

Count the spaces, not the tick marks: from 0 to 1 there are eight steps, so each is an eighth, and the point stands at the fifth. Counting the marks instead of the gaps is the classic number-line slip, and it yields a wrong answer every time. (Writing it as 5/8 exactly is right; the fraction is already in simplest form, since 5 is prime and doesn't divide 8.)

4 B: Bar B.

Bar B is cut into six equal pieces with three shaded, and 3/6 is another name for one half — the shading reaches the middle. A) is cut into two pieces, which looks like halves until you notice the pieces aren't equal: two pieces do not make halves unless they match. C) shades one of four, which is a quarter. This question is the figure from The Guide, asked backward.

5 12

5 becomes 20 by multiplying by 4, so the top takes the same treatment: 3 × 4 = 12. The error to watch for is adding 15 to both, giving 18/20 — but re-cutting is multiplication, not addition. If you add to both, the pieces don't change size evenly and the amount shifts.

6 3/8

Both end in 5 or 0, so both divide by 5: 15 ÷ 5 = 3 and 40 ÷ 5 = 8. Now 3 and 8 share only 1, so it's finished. Note where this landed: the same fraction as question 1, arrived at from a very different-looking place.

7 2/3

Both even, so divide by 2: 12/18. Both even again: 6/9. Digit sums 6 and 9 both divide by 3: 2/3. Or spot the 12 at once and land there in one step. Either route is correct, and both stop in the same place, because simplest form is unique. If you stopped at 6/9 or 12/18, the work wasn't finished: keep going until the only shared factor is 1.

8 A: 1¾.

The bar means divide: 7 ÷ 4 = 1 R 3, so one whole and 3 fourths left over. B) 3¼ swapped the quotient and the remainder. C) 7¼ kept the 7 whole and invented a quarter. D) simply flipped the fraction over. Check it against the number line in The Guide: 7/4 and 1¾ sit on the very same tick.

9 $21.75, or 21¾ dollars

The three leftover singles are shared by 4, giving each person 3/4 of a dollar on top of the 21 they already hold. Three quarters of a dollar is 75 cents, which is why money is such good practice for fractions: every price tag is a fraction of a dollar already written in another alphabet. This is the fourth fate Quiz 5 promised — the remainder itself divided, so nothing is left on the counter.

10 3/4 of the shift

6 out of 8 is 6/8, and both are even, so divide by 2: 3/4. The whole is the whole shift, all 8 hours, not the 2 hours remaining. And notice something this quiz has been doing quietly: 3/4 has now appeared as slices, as an address on the line, as 18/24 simplified, as the leftover from a division, and as most of a workday. One amount, many disguises.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2, 4	What the two numbers mean	Bottom names the size, top counts them; and the pieces must be equal or it isn't a fraction at all.
3	Fractions on the number line	Count the spaces between marks, never the marks themselves.
5	Equivalence	Re-cutting is multiplication on both numbers, never addition.
6, 7, 10	Simplifying	Quiz 7's tests, applied to both numbers; stop only when the shared factor is 1.
8, 9	Improper fractions and mixed numbers	The bar means divide; the remainder becomes the numerator.

Eight or more right: fractions are open to you — on to Quiz 9. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Fold paper into halves, then fourths, then eighths with your teacher, and watch one amount take three names. Then come back to these same ten.

The Company · An Interlude

Fellow travelers

A different masthead today: no cut-out supporters, just one seahorse in its own water, given the whole field. The title sits where the seahorse is looking. Below, that same soloist seen entire.

A golden seahorse hovering upright among eelgrass over white sand, tail curled, with a second seahorse blurred behind — The soloist, holding station among the eelgrass. Note the second seahorse gripping a branch behind, out of focus and out of the argument: even a solo has a section resting somewhere in the room.

A mottled seahorse with a deep pale belly holding station against dark reef growth — And a returning face from Quiz 4, since the parts of a whole is today's business: patient, upright, unhurried.

Prints only this gallery, full size. The main print button stays photograph-free to spare your ink.