Seahorse Series · Quiz 5 · Division

The Guide

Before you begin

Division is the last of the four operations, and it asks two everyday questions with one piece of arithmetic. One is the sharing question: 84 dollars among 6 people, how much each? The other is the measuring question: how many 6-dollar tickets can 84 dollars buy? The arithmetic is the same either way, and every word problem asks one of the two. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

Multiplication, run backward

Every division is a multiplication question read in reverse: 84 ÷ 6 asks 6 times what makes 84? The times table (the toolkit, in Quiz 4) runs in both directions, and every fact is really a family of three: 6 × 14 = 84, so 84 ÷ 6 = 14, and 84 ÷ 14 = 6. One rectangle, three questions.

The rectangle with a missing side

The rectangle, Quiz 4's picture of multiplication, returns with one number hidden. If a rectangle holds 84 square units and one side is 6, division finds the other side:

Division is the rectangle with a missing side. Multiplication built the floor; division measures the wall.

Long division: dealing out, biggest bills first

The written method is a dealer at the cash register, paying out a sum in equal shares — largest bills first. Deal the hundreds. Whatever won't deal evenly breaks into tens and joins the tens (the same trade as breaking a ten in subtraction, in Quiz 3). Then deal the tens; whatever is left breaks into ones; then deal the ones. Each place of the answer records how many that place dealt.

Picture it at the cash register: pay out 87 dollars to 4 people. The 8 tens deal 2 tens each: 8 gone, none left. The 7 singles deal 1 each: 4 gone, and 3 singles sit on the counter that no one can take without breaking them into coins. That leftover is the remainder, written R 3: 87 ÷ 4 = 21 R 3. (Breaking those last singles into fair shares of a dollar is a real move, but it belongs to fractions, a few quizzes from now.)

Two laws of the remainder

First: a remainder is always smaller than the divisor, the number you are dividing by — if 4 or more singles were sitting on the counter, another round could still be dealt, so keep dealing. Second: the round trip proves any division — multiply back and add the leftover, and you must land exactly where you began: 4 × 21 + 3 = 87. If the trip misses home, the deal was miscounted.

Zeros in the quotient

Sometimes a place deals out nothing — and Quiz 1's oldest law holds: the zero must still be written. In 618 ÷ 6, the hundreds deal 1, the lone ten deals 0 sixes, the zero takes its seat, and 18 ones deal 3: the answer is 103, not 13. A skipped zero collapses the whole number.

The guard, with compatible numbers

Division's estimate uses compatible numbers: slide the dividend (the number being divided) to a close neighbor that the divisor divides cleanly. For 251 ÷ 8, slide to 240: about 30. The guard — the estimate you make before trusting an answer — speaks first, as ever.

The three fates of a remainder

Here is the heart of this quiz. The arithmetic hands you a quotient and a remainder — but the situation decides what they mean. A remainder meets one of three fates: it rounds the answer up (people still need a ride, so another car is needed); it gets dropped (only full boxes count); or it is the answer itself (how many are left over?). Same division, three different answers. The question decides — never the arithmetic alone.

Worked Examples

Three to study before you start

Example 1 · The family, the rectangle

84 ÷ 6 asks: 6 times what makes 84? The toolkit answers sideways: 6 × 14 = 84, so 84 ÷ 6 = 14: the figure's missing side, found. And the third family member comes free: 84 ÷ 14 = 6.

Example 2 · The written deal

87 ÷ 4, dealt largest places first:

	2	1	R 3
4	8	7
	−8
	0	7
		−4
		3

The 8 tens deal 2 apiece — write 2 over the tens, subtract the 8 dealt, nothing left. The 7 comes down; it deals 1 apiece — write 1, subtract 4, and 3 sits on the counter: 21 R 3. Round trip: 4 × 21 + 3 = 87 — home.

Example 3 · A longer deal, guarded

1,258 ÷ 5. Guard first: slide to 1,250 — about 250. Now deal: the 1 thousand can't deal to 5 alone, so it breaks into the hundreds: 12 hundreds deal 2 apiece (10 dealt, 2 left). The 2 hundreds break into the tens: 25 tens deal 5 apiece, clean. The 8 ones deal 1 apiece, 3 left. Answer: 251 R 3, right beside the guard's 250. Round trip: 5 × 251 + 3 = 1,258 — home again.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Write your answers on the lines. Questions 5 and 8 are multiple choice — choose the one best answer.

Divide: 72 ÷ 8
A rectangle holds 91 square feet and one side is 7 feet. How long is the other side?
Divide: 96 ÷ 4
Divide, give the remainder, and prove it with the round trip: 94 ÷ 6
Divide: 618 ÷ 6
- A) 13
- B) 130
- C) 103
- D) 1,003
Divide: 2,438 ÷ 7

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

Take two minutes with the leafy and weedy seadragons, filmed by Wired.

Estimate with compatible numbers: 251 ÷ 8
Fifty-eight members of the union local are carpooling to Albany for a hearing. Each car seats 6. How many cars does the local need?
- A) 9
- B) 9 R 4
- C) 10
- D) 4
A bakery packs muffins 6 to a box. The morning bake is 58 muffins. How many full boxes can go out for sale?
Same bakery, same morning: after the boxes are filled, how many muffins are left on the tray for the bakers?

Answer Key

Check your work

Open the key — after you've finished all ten

1 9

The toolkit read sideways: 8 × 9 = 72. If this wobbled, the work is the facts themselves, not the method — the table in Quiz 4's toolkit runs both directions.

2 13 feet

7 × ? = 91. This one goes past the edge of the times table on purpose: 7 × 10 = 70, and the remaining 21 is 7 × 3 — so 13. Division is the rectangle's missing side, and big sides are found in pieces.

3 24

Deal the 9 tens: 2 apiece, 8 dealt, 1 ten left. It breaks into the ones: 16 ones deal 4 apiece, clean. Round trip: 4 × 24 = 96.

4 15 R 4 · 6 × 15 + 4 = 94

The 9 tens deal 1 apiece; 3 tens break down to join the 4 ones; 34 ones deal 5 apiece with 4 on the counter. Note the first law: if your remainder was 6 or more, another round could still be dealt — a remainder must always be smaller than the divisor.

5 C: 103.

The hundreds deal 1; the lone ten deals zero sixes, and that zero must be written; then 18 ones deal 3. A) dropped the zero and collapsed the number. B) parked the zero at the end, ten times too big. D) invented a place that doesn't exist. Round trip: 6 × 103 = 618.

6 348 R 2

Guard: between 2,100 ÷ 7 = 300 and 2,800 ÷ 7 = 400. The deal: 24 hundreds give 3 apiece (21 dealt, 3 left); 33 tens give 4 apiece (28 dealt, 5 left); 58 ones give 8 apiece (56 dealt, 2 on the counter). Round trip: 7 × 348 + 2 = 2,438 — home.

7 About 30

Slide 251 to its compatible neighbor 240, and 240 ÷ 8 = 30. (The exact answer is 31 R 3 — the estimate landed a step away, in seconds.) Any sound compatible pair earns full credit; the skill is choosing the divisor's friendly neighbor at all.

8 C: 10 cars.

58 ÷ 6 = 9 R 4. Nine cars seat 54 — and four members are still standing on the curb. The local doesn't leave people on the curb, so the remainder forces a tenth car: the remainder rounds the answer up. A) strands four people. B) is honest arithmetic but no one has ever driven "9 R 4 cars." D) counts the stranded instead of the cars. Notice something about this question's wrong answers — each one is the right answer to a different question. That is the entire lesson of the next two items.

9 9 full boxes

Same division, 58 ÷ 6 = 9 R 4, but now the four leftover muffins can't fill a box, so this answer drops the remainder. Only whole boxes go out for sale.

10 4 muffins

And now the remainder is the answer. Stand back and look at what just happened: questions 8, 9, and 10 were all one computation, 58 ÷ 6 = 9 R 4, and it produced three different answers: 10, then 9, then 4. The arithmetic never changed; the questions did. That is the deepest fact in this quiz: the situation decides what the numbers mean. (A fourth fate — cutting the leftovers themselves into equal shares — waits in the fractions quizzes, a few quizzes from now.)

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Multiplication run backward	The toolkit sideways; the rectangle's missing side.
3, 4, 6	The long-division deal	Largest places first; leftovers trade down; a remainder stays smaller than the divisor; the round trip must land home.
5	Zeros in the quotient	Every place gets a digit — even when a place deals out nothing.
7	Estimation	Compatible numbers: slide to the divisor's friendly neighbor.
8, 9, 10	What the remainder means	The three fates — up, dropped, or the answer itself. The question decides, never the arithmetic alone.

Eight or more right: the deal is yours — on to Quiz 6. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news — we've found the right counter to work at. Deal real coins or dried beans with your teacher — hundreds, tens, ones, biggest first — then come back to these same ten.

The Company · An Interlude

Fellow travelers

The masthead trio holds its formation above. Down here, two photographs. Quiz 4's gallery said that the opal seahorse at the masthead is an artist's invention; both of these are the genuine article, and one of them proves a point about it.

An orange seahorse wrapped among orange coral branches, its body textured with fine bumps, photograph credited in-frame to Zylstra and Rothenfluh — An orange seahorse at home in orange coral — photograph credited in-frame to Zylstra & Rothenfluh, whose names we keep visible. Camouflage as a way of belonging.

A weedy seadragon photographed in black and white, rising through dark water with light falling from above — The weedy seadragon in silver and black — the family holds its bearing even with the color turned off. Structure, not palette, is the truth of a body; that is also how you spot the fakes.

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