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Number · Number and Numeration Systems · Lesson 2

Comparing and Ordering Numbers beyond One Billion

Which region has more people? Which year did Ghana earn more from cocoa? Which bank holds more money? To answer questions like these you must compare very big numbers quickly and correctly. In this lesson you will compare and order numbers beyond one billion, use the symbols >, < and =, skip count in 25s, 50s and 250s, and find numbers that are more or less than a given number.

NaCCA B7.1.1.1.2 ⏱ about 45 min

By the end you will be able to

  • ✔Compare two whole numbers greater than 1,000,000,000 using place value and write the result with >, < or =.
  • ✔Arrange a set of large numbers in ascending and descending order.
  • ✔Skip count forwards and backwards in 25s, 50s and 250s from 1,000.
  • ✔Find a number that is a given amount (e.g. 500,000) more than or less than a large number.
  • ✔Read and write numbers in given positions on a number chart.

Key words

Greater than ( > )
Shows the first number is bigger, e.g. 5,223,487,637 > 5,113,487,637.
Less than ( < )
Shows the first number is smaller, e.g. 999,999,999 < 1,000,000,000.
Equal to ( = )
Shows both numbers have exactly the same value.
Ascending order
Arranged from the smallest to the largest.
Descending order
Arranged from the largest to the smallest.
Skip counting
Counting forwards or backwards by adding or subtracting the same number each time, e.g. in 250s.
Curriculum reference

Content standard: B7.1.1.1 Demonstrate understanding and the use of place value for expressing quantities recorded as base ten numerals as well as rounding these to given decimal places and significant figures.

Indicator: Compare and order whole numbers more than 1,000,000,000 and represent the comparison using ">, <, or =".

Starter

Who sold more cocoa?

Two cocoa buying companies in Kumasi report their sales for the year. Company A says GH₵5,223,487,637 and Company B says GH₵5,113,487,637. The numbers look almost the same — same length, even the same last seven digits — yet one company sold more than one hundred million cedis' worth extra! How can you spot the bigger number in a few seconds without adding or subtracting anything?

1. Comparing two big numbers using place value

When you compare two numbers you are asking: which one is bigger, or are they the same? For small numbers like 47 and 74 you can see the answer at once. For numbers in the billions we need a careful method, and that method is built on place value — the idea from the last lesson that each place is worth 10 times the place on its right.

  1. 1Count the digits. Write both numbers with commas. The number with more digits is greater. For example 1,000,000,000 (10 digits) > 999,999,999 (9 digits).
  2. 2If they have the same number of digits, line them up so that billions sit under billions, millions under millions, and so on.
  3. 3Compare digit by digit from the LEFT (the biggest place). Move right until you find the first place where the digits are different.
  4. 4The number with the bigger digit in that place is the greater number. Everything to the right of that place does not matter any more.
  5. 5If every digit is the same, the numbers are equal.
Lining up 5,223,487,637 and 5,113,487,637 by place value
NumberBillionsHundred millionsTen millionsMillionsThousands groupOnes group
5,223,487,6375223487637
5,113,487,6375113487637

Both numbers have 5 billions, so we move right. In the hundred-millions place the first number has 2 and the second has 1. Since 2 > 1, we stop: 5,223,487,637 > 5,113,487,637. We can also say 5,113,487,637 is less than 5,223,487,637, written 5,113,487,637 < 5,223,487,637. Both sentences say the same thing.

▶ On a number line, numbers grow from left to right. Notice that the number further to the right is always the greater one.

2. Putting numbers in order

Ordering is just comparing many times. To put numbers in ascending order (smallest first), find the smallest number, write it down, then find the smallest of those left, and so on. Descending order is the same list turned round (largest first).

A good way is to compare all the numbers place by place at the same time. Suppose the cocoa earnings (in cedis) of four companies are 2,405,300,000; 2,450,030,000; 2,045,900,000 and 2,405,030,000. All have 10 digits and all start with 2 billion. Now look at the hundred-millions digit: 4, 4, 0, 4. The one with 0 (2,045,900,000) is the smallest. For the three with 4, look at the ten-millions digit: 0, 5, 0 — so 2,450,030,000 is the largest. For 2,405,300,000 and 2,405,030,000 the first difference is in the hundred-thousands place: 3 > 0.

The four numbers in ascending order
PositionNumberReason
1st (smallest)2,045,900,0000 hundred millions
2nd2,405,030,0000 hundred thousands
3rd2,405,300,0003 hundred thousands
4th (largest)2,450,030,0005 ten millions

So in ascending order: 2,045,900,000 < 2,405,030,000 < 2,405,300,000 < 2,450,030,000. In descending order we simply reverse it: 2,450,030,000 > 2,405,300,000 > 2,405,030,000 > 2,045,900,000.

3. Skip counting in 25s, 50s and 250s

Skip counting helps you see how numbers grow by equal amounts. It is what a trotro mate does when he collects GH₵25 at a time, or what a bank teller does when counting GH₵50 notes. To count forwards we keep adding the same number; to count backwards we keep subtracting it.

Skip counting from 1,000
Count inForwardsBackwards
25s1,000, 1,025, 1,050, 1,075, 1,100, 1,1251,000, 975, 950, 925, 900, 875
50s1,000, 1,050, 1,100, 1,150, 1,200, 1,2501,000, 950, 900, 850, 800, 750
250s1,000, 1,250, 1,500, 1,750, 2,000, 2,2501,000, 750, 500, 250, 0

▶ Every jump is the same size. Notice that four jumps of 250 make 1,000.

4. More than, less than and number charts

To find a number that is 500,000 more than a given number, add 500,000. To find a number that is 500,000 less than it, subtract 500,000. Because 500,000 is 5 hundred-thousands, usually only the hundred-thousands digit changes (unless you need to carry or borrow).

  • 1,295,800,000 is 500,000 more than 1,295,300,000, because 1,295,300,000 + 500,000 = 1,295,800,000.
  • 1,295,300,000 is 500,000 less than 1,295,800,000, because 1,295,800,000 − 500,000 = 1,295,300,000.
  • Watch the carry: 500,000 more than 3,487,600,000 is 3,488,100,000 (6 + 5 = 11 hundred-thousands, so carry 1 million).

A number chart is a grid where numbers follow a rule across each row and down each column. In the chart below each number is 500,000 more than the one on its left, and each number is 2,000,000 more than the one above it.

A number chart (rule: +500,000 across, +2,000,000 down)
Column 1Column 2Column 3Column 4
187,500687,5001,187,5001,687,500
2,187,5002,687,5003,187,5003,687,500
4,187,5004,687,5005,187,5005,687,500

Which number is on the right of 3,187,500? Move one box right: 3,687,500. In words: three million, six hundred and eighty-seven thousand, five hundred. Once you know the rule, you can also fill in missing boxes and extend the chart, for example the next row starts with 6,187,500.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Equal numbers

Insert >, < or = to make the statement true: 1,300,850,700 .... 1,300,850,700
Show the solution
  1. Step 1Both numbers have 10 digits.
  2. Step 2Compare from the left: billions 1 and 1, millions group 300 and 300, thousands group 850 and 850, ones group 700 and 700.
  3. Step 3Every digit is the same, so the numbers are equal.

Answer: 1,300,850,700 = 1,300,850,700

Example 2

Finding the first different digit

Insert >, < or =: 5,223,487,637 .... 5,113,487,637
Show the solution
  1. Step 1Both have 10 digits, so line them up.
  2. Step 2Billions: 5 and 5 — the same, move right.
  3. Step 3Hundred millions: 2 and 1 — different! 2 > 1.
  4. Step 4So the first number is greater. The mouth of the symbol opens towards 5,223,487,637.

Answer: 5,223,487,637 > 5,113,487,637

Example 3

Ordering four numbers

Arrange in ascending order: 2,405,300,000; 2,450,030,000; 2,045,900,000; 2,405,030,000.
Show the solution
  1. Step 1All have 10 digits and 2 billions.
  2. Step 2Hundred millions: 4, 4, 0, 4 → 2,045,900,000 is the smallest.
  3. Step 3Ten millions of the rest: 0, 5, 0 → 2,450,030,000 is the largest.
  4. Step 4Compare 2,405,300,000 and 2,405,030,000: same until the hundred-thousands place, where 3 > 0, so 2,405,030,000 is smaller.
  5. Step 5Write them smallest to largest.

Answer: 2,045,900,000; 2,405,030,000; 2,405,300,000; 2,450,030,000

Example 4

Skip counting backwards

Count backwards in 250s from 1,000 and write the first four numbers after 1,000.

▶ Each backward jump subtracts 250.

Show the solution
  1. Step 11,000 − 250 = 750.
  2. Step 2750 − 250 = 500.
  3. Step 3500 − 250 = 250.
  4. Step 4250 − 250 = 0.

Answer: 750, 500, 250, 0

Example 5

More than, less than and the number chart

(a) Which number is 500,000 less than 1,295,800,000? (b) In the number chart, which number is on the right of 3,187,500? Write it in words.
Show the solution
  1. Step 1(a) Subtract: 1,295,800,000 − 500,000. Only the hundred-thousands digit changes: 8 − 5 = 3.
  2. Step 2So the answer is 1,295,300,000.
  3. Step 3(b) The rule across the chart is +500,000. 3,187,500 + 500,000 = 3,687,500.
  4. Step 4Read the groups: 3 | 687 | 500 → three million, six hundred and eighty-seven thousand, five hundred.

Answer: (a) 1,295,300,000 (b) 3,687,500 — three million, six hundred and eighty-seven thousand, five hundred

Summary

  • The number with more digits is greater. If the digit counts are equal, compare place by place from the LEFT.
  • The first place where the digits differ decides which number is greater; use >, < or = to record it.
  • Ascending order is smallest to largest; descending order is largest to smallest.
  • Skip counting means adding (forwards) or subtracting (backwards) the same amount each time, e.g. 25, 50 or 250.
  • 500,000 more means add 500,000; 500,000 less means subtract 500,000 — watch for carrying and borrowing.
  • In a number chart, find the rule across and down, then use it to find any position.

Practice: 15 questions

5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.

You can practise without an account, but sign up free or sign in to save your progress.

1 easy
Choose the symbol that makes this true: 4,560,000,000 .... 4,650,000,000
2 easy
Count forwards in 25s: 1,000, 1,025, 1,050, ____. What is the next number?
3 easy
What number is 500,000 more than 1,295,300,000?
4 easy
Which number is the greatest?
5 easy
Write the symbol >, < or = that makes this true: 1,300,850,700 .... 1,300,850,700
6 medium
Count backwards in 250s: 1,000, 750, 500, ____. What is the next number?
7 medium word problem
A cocoa buying company recorded these earnings in four years: Year A GH₵9,512,000,000; Year B GH₵9,215,000,000; Year C GH₵9,521,000,000; Year D GH₵9,125,000,000. Which list is in ascending order?
8 medium word problem
A bank in Accra had GH₵2,750,400,000 in its vault. It then received GH₵500,000 more. How much does it have now?
9 medium
Use the number chart in the lesson (rows: 187,500 · 687,500 · 1,187,500 · 1,687,500 / 2,187,500 · 2,687,500 · 3,187,500 · 3,687,500 / …). Which number is directly below 1,687,500?
10 medium
Which statement is TRUE?
11 hard word problem
Akosua's company recorded its mobile money transfers for three months: January GH₵1,295,300,000; February GH₵1,295,800,000; March GH₵1,259,800,000.

  1. (a) In which month were the transfers greatest?

  2. (b) How much more (in cedis) was February than January?

  3. (c) Write the symbol >, < or = to compare January .... March.

12 hard word problem
Kwame starts at 1,000 and counts forwards in 250s. 1,000 is his 1st number, 1,250 is his 2nd number. What is his 10th number?
13 hard
In the number chart of the lesson, the number on the right of 3,187,500 is 3,687,500. Write 3,687,500 in words.
14 hard
Using each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once, write the smallest 10-digit number.
15 hard
Yaw says, "999,999,999 > 1,000,000,000 because 9 is bigger than 1." Which reply is correct?