Seahorse Series · Quiz 2 · Comparing, Ordering, and Rounding

The Guide

Before you begin

This quiz is about three everyday acts: deciding which of two numbers is bigger, putting several numbers in order, and trading an exact number for a nearby round one. All three use place value, the subject of Quiz 1. If you did that quiz, you have the tool already; if not, everything you need is below. As in every quiz of this series: nothing is a trick, no calculator is needed, and whatever the quiz shows is useful news.

Comparing: count first, then walk

First, count the digits. For whole numbers, more digits wins — always. A six-digit number beats any five-digit number, because even the smallest six-digit number, 100,000, is bigger than the biggest five-digit number, 99,999. If the counts differ, you're done.

If the counts match, walk. Start at the left and compare place by place, together. The first place where the digits differ decides everything — and nothing after it gets a vote. Here is 508,214 against 508,124:

5	0	8	,	2	1	4
5	0	8	,	1	2	4

The figure is wider than your screen — slide it sideways with your finger, or turn your phone.

Five ties five, zero ties zero, eight ties eight. Then 2 stands against 1 — and the walk is over: 508,214 is bigger. The faded digits after the decision never mattered.

Why does the first difference get the only vote? Because the places to the right, even at their strongest, can't make up the difference. At the hundreds place, 2 beats 1 by a full hundred — and the most the tens and ones can ever add up to is 99. It's like alphabetizing names: Sanders comes before Santiago because of one letter, d against t, and no letter after that can change the order.

The two symbols

The signs < and > are wedges: the wide open side always faces the bigger number, and the point pokes the smaller one. So 98,764 < 100,002 and 100,002 > 98,764 state the same fact from its two sides.

Ordering is comparing, repeated

To put several numbers in order, find the smallest by the count-then-walk method, set it down, and repeat with what's left. A practical tip from Quiz 1: stack the numbers so their places line up — ones under ones, tens under tens — and the walk becomes something your eye can do at a glance.

The number line: every number has an address

Numbers live on a line, like houses along a road. Bigger is to the right, smaller is to the left — that single fact is the whole geometry of it. When you order numbers, you are really just reporting the order of their houses along the road.

A stretch of the number road. Not every tick wears a label — the one between 340 and 360 is 350, whether it says so or not.

Rounding: the nearest landmark

Round numbers — the ones ending in zeros — are the landmarks of the road: the tens, the hundreds, the thousands stand at even spacing like streetlights. To round a number is to answer one question: which landmark am I closest to? Take 368, rounded to the nearest ten. It stands between the landmarks 360 and 370:

368 stands two steps from 370 and eight steps from 360. It rounds to 370 — nearness decides.

The shortcut rule, once the picture is yours: find the place you're rounding to, then look one door to the right. If that digit is 5 or more, round up; if it's 4 or less, stay. (Exactly halfway, as with 365 standing between 360 and 370, is nearest to neither, so custom settles it: round up.) One look, one decision. Never round in stages; a number rounded twice can drift right past its true landmark.

When the nine rolls over

Sometimes rounding up crosses a bigger boundary. Round 5,972 to the nearest hundred: its landmarks are 5,900 and 6,000, and it stands only 28 steps from the top one.

5,972 rounds to 6,000 — and the 9 rolls over like a car odometer: 59 hundreds become 60 hundreds.

Now the same thing in digits. The tens digit is 7, so the hundreds place rounds up. Rounding a 9 up makes it roll over into the next place, exactly like an odometer turning from 59,999 to 60,000 miles. The answer is 6,000, zeros and all.

Estimation: rounding with a purpose

Rounding earns its keep as a habit: before trusting any answer — yours, a cashier's, a machine's — round the ingredients and ask what neighborhood the true answer should live in. An estimate doesn't do the math. It stands guard over it.

Worked Examples

Three to study before you start

Example 1 · Comparing, both rules

Compare 63,940 and 9,876: count the digits — five against four — and it's over: 63,940 is bigger, no walking needed. Now 271,536 against 271,498: same count, so walk. 2 against 2, 7 against 7, 1 against 1, then 5 against 4 — done: 271,536 > 271,498.

Example 2 · Rounding — one look, one decision

Round 7,384 to the nearest thousand. Find the thousands digit: 7. Look one door to the right: the hundreds digit is 3 — four or less, so stay. Answer: 7,000. Notice what we did not do: we never rounded 384 up to 400 first and then asked again. One look at one digit; everything to its right becomes zeros and falls silent.

Example 3 · The estimate stands guard

Marcus adds 4,286 + 3,910 and writes down 5,196. Check the neighborhood: 4,286 is about 4,000, and 3,910 is about 4,000, so the true answer should live near 8,000. An answer of 5,196 is in the wrong part of town entirely — time to re-add. (He'll find 8,196; a digit had slipped.) The estimate caught in five seconds what re-checking might have missed.

The Quiz · Ten Questions

Now you

Work without a calculator. Write your answers on the lines. Questions 1 and 8 are multiple choice — choose the one best answer.

Which symbol makes this true? 98,764 ___ 100,002
- A) <
- B) >
- C) =
Write < or > to make this true: 508,214 ___ 508,124
Put these in order from least to greatest: 42,506 · 42,065 · 42,560 · 42,056
The gold point on the number line marks a number.

a) Between which two tens does it stand?

b) Which of those two tens is it nearer to?
Round 4,286 to the nearest hundred.
Round 34,462 to the nearest thousand.
Round 5,972 to the nearest hundred.
Which of these numbers rounds to 2,500 when rounded to the nearest hundred?
- A) 2,449
- B) 2,463
- C) 2,554
- D) 2,571
At the hardware store, Marcus buys a drill for $189, a ladder for $312, and a shop vacuum for $67. Round each price to the nearest hundred and estimate the total.
A stadium held 47,832 fans. A reporter writes, "about 48,000 attended." To which place did the reporter round — the nearest hundred, the nearest thousand, or the nearest ten-thousand?

Answer Key

Check your work

Open the key — after you've finished all ten

1 A: 98,764 < 100,002.

Count before you walk: five digits against six. If you chose B, you likely compared the front digits, 9 against 1, without counting first. But a five-digit number can never beat a six-digit one: even 100,000, the smallest of them, tops 99,999, the biggest five-digit number there is.

2 508,214 > 508,124

Same count, so walk: 5 against 5, 0 against 0, 8 against 8, then 2 against 1 — decided at the hundreds place. The 14 and 24 at the end never got a vote; the most they could swing is 99, and the decision was worth a hundred.

3 42,056 · 42,065 · 42,506 · 42,560

All four begin 42, so every decision lives in the last three places. Stack them and walk: 056 before 065 (tens: 5 under 6), both before 506 and 560 (hundreds: 0 under 5), and 506 before 560 (tens: 0 under 6). If any pair tripped you, it was almost certainly a swapped-digit pair — this question is built entirely out of them.

4 a) Between 340 and 350. b) Nearer to 340.

The tick to the right of 340 wears no label, but the ticks march by tens, so it is 350 all the same — scales on the GED often label only some of their ticks and expect you to count the rest. The point sits about two steps past 340 and eight before 350: it is nearer to 340.

5 4,300

Hundreds digit: 2. One door right: the tens digit is 8 — five or more, so up: 4,300. On the line, 4,286 stands only 14 steps from 4,300 and 86 from 4,200.

6 34,000

Thousands digit: 4. One door right: the hundreds digit is 4 — stay: 34,000. If you answered 35,000, you probably rounded in stages: 462 up to 500, then 34,500 up to 35,000. That's the drift the Guide warned about: each small rounding was legal, but the chain carried the number right past its true landmark. 34,462 is 462 steps from 34,000 and 538 from 35,000 — so it is not a close call. One look, one decision.

7 6,000

Hundreds digit: 9. One door right: 7 — round up, and the 9 rolls over, carrying into the thousands like an odometer: 59 hundreds become 60 hundreds. If you wrote 5,1000 or 5,000, the carry is the piece to revisit — it's the same carrying that Quiz 3 will make routine.

8 B: 2,463.

Check each: 2,463 has tens digit 6, so it rounds up to 2,500. A) 2,449 is only 51 away from 2,500 — but nearness is measured to the closest landmark, and 2,400 is closer still, at 49. C) 2,554 rounds up to 2,600 (tens digit 5), and D) 2,571 rounds to 2,600 as well. The lesson inside this one: a number can be quite close to a landmark and still belong to a different one.

9 About $600

$189 rounds to $200, $312 rounds to $300, and $67 rounds to $100 — so the estimate is $200 + $300 + $100 = $600. The true total is $568; the estimate runs a little high because two of the three prices rounded up. That's the habit in action: before the register finishes, Marcus already knows what neighborhood the total should live in.

10 The nearest thousand.

Test each place: to the nearest hundred, 47,832 becomes 47,800; to the nearest ten-thousand, it becomes 50,000. Only rounding to the nearest thousand produces 48,000 (hundreds digit 8, so the thousands round up). Reading the news this way — asking what did they round, and how far? — is estimation running in reverse.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2, 3	Comparing and ordering	Reread "Count first, then walk." If the walk itself wobbles, the place-value chart in Quiz 1 is the foundation to revisit.
4	Reading the number line	Practice counting unlabeled ticks — the road picture in The Guide, and any GED graph, depends on it.
5, 6, 7	Rounding by the rule	One look, one decision — and when a 9 rounds up, let it roll over like the odometer.
8, 10	What rounding means	Go back to the landmark pictures — nearness decides, and every number belongs to exactly one landmark per place.
9	Estimation as a habit	Round first, then add — and let the estimate stand guard over the answer.

Eight or more right: comparing and rounding are solid — move on to Quiz 3. Five to seven: review the rows flagged above and retake this quiz in a few days. Fewer than five: good news — we've found the right place to work. Walk through the number-line pictures slowly with your teacher, then come back to these same ten questions.