Measurement of Money

Coin Identification and Values

Money helps us buy things. In this lesson, we will learn how to identify coins and know how much each coin is worth.

We will practice with four common coins: the penny, nickel, dime, and quarter.

Important idea: A coin’s size does not always tell its value. Some small coins are worth more than bigger coins.

Meet the coins

• Penny = \(1\) cent
• Nickel = \(5\) cents
• Dime = \(10\) cents
• Quarter = \(25\) cents

You can write cents with the cent sign or the word cents, like this: \(1\) cent, \(5\) cents, \(10\) cents, and \(25\) cents.

How to identify each coin

Penny

• Worth \(1\) cent
• Usually looks copper or brown
• It is one of the smallest-value coins

When you see a penny, think: "Penny = 1".

Nickel

• Worth \(5\) cents
• Silver-colored
• Bigger than a penny and a dime

When you see a nickel, think: "Nickel = 5".

Dime

• Worth \(10\) cents
• Silver-colored
• Smallest of these four coins

The dime is small, but it is worth more than a penny and a nickel. When you see a dime, think: "Dime = 10".

Quarter

• Worth \(25\) cents
• Silver-colored
• Usually the biggest of these four coins

When you see a quarter, think: "Quarter = 25".

A good way to remember

• Penny \(= 1\)
• Nickel \(= 5\)
• Dime \(= 10\)
• Quarter \(= 25\)

You can say them like this:

"1, 5, 10, 25"

Be careful!

• The dime is smaller than the nickel, but the dime is worth more.
• The penny is the only coin here that is copper-colored.
• The quarter is worth the most of these four coins.

Worked Example 1: Identify one coin

Question: What is the value of a nickel?

Step 1: Remember the coin values.

Penny \(= 1\), Nickel \(= 5\), Dime \(= 10\), Quarter \(= 25\).

Step 2: Find nickel.

A nickel is worth \(5\) cents.

Answer: A nickel is worth \(5\) cents.

Worked Example 2: Compare two coins

Question: Which coin is worth more, a penny or a dime?

Step 1: Say each value.

• Penny = \(1\) cent
• Dime = \(10\) cents

Step 2: Compare the numbers.

Since \(10 > 1\), the dime is worth more.

Answer: A dime is worth more than a penny.

Worked Example 3: Identify coins in a group

Question: Name the values of these coins: penny, quarter, dime.

Step 1: Match each coin to its value.

• Penny = \(1\) cent
• Quarter = \(25\) cents
• Dime = \(10\) cents

Answer: The values are \(1\) cent, \(25\) cents, and \(10\) cents.

Worked Example 4: Put coins in order by value

Question: Put these coins in order from least value to greatest value: quarter, penny, nickel, dime.

Step 1: Write each value.

• Quarter = \(25\)
• Penny = \(1\)
• Nickel = \(5\)
• Dime = \(10\)

Step 2: Put the values in order.

$$1, 5, 10, 25$$

Step 3: Match the order back to the coins.

Penny, Nickel, Dime, Quarter

Answer: From least to greatest value: penny, nickel, dime, quarter.

Tips for remembering coin values

• Penny starts with P, and penny is the smallest value: \(1\).
• Nickel sounds like five is a friendly partner to remember.
• Dime is tiny, but it is worth \(10\).
• Quarter means one part of four equal parts of a dollar, and it is worth \(25\) cents.

Quick check

1. How much is a penny worth?
2. Which coin is worth \(10\) cents?
3. Which coin is worth the most: nickel or quarter?
4. Which coin is the smallest but worth more than a nickel?

Quick check answers

1. A penny is worth \(1\) cent.
2. A dime is worth \(10\) cents.
3. A quarter is worth more.
4. A dime.

Summary

There are four important coins to know: penny, nickel, dime, and quarter.

Their values are:

• Penny = \(1\) cent
• Nickel = \(5\) cents
• Dime = \(10\) cents
• Quarter = \(25\) cents

Remember: the dime is small, but it is worth more than a nickel. Knowing coin names and values helps you count money and solve money problems.

Counting Mixed Coin Collections means finding the total value of different coins together.

When we count mixed coins, it helps to put the coins in order from highest value to lowest value first. This makes counting easier and helps us not miss any coins.

We will use these coins:

• Quarter = 25 cents
• Dime = 10 cents
• Nickel = 5 cents
• Penny = 1 cent

A good order to remember is:

Quarter, dime, nickel, penny

That means:

$$25,\ 10,\ 5,\ 1$$

When coins are mixed up, do these steps:

1. Name the coins.
2. Put them in order from greatest value to least value.
3. Count on from the biggest coins first.
4. Say the total in cents.

Let’s learn how to count on.

If you start with a quarter, you start at \(25\).

If you add a dime, count on 10 more: \(25 + 10 = 35\).

If you add a nickel, count on 5 more: \(35 + 5 = 40\).

If you add pennies, count on by ones: \(40, 41, 42, 43\).

This is faster than counting every coin as 1.

Why do we start with the biggest coins?

Starting with the biggest coins helps our brains keep track more easily. It is simpler to count \(25\), then \(10\), then \(5\), then \(1\) than to jump around.

Think of it like cleaning up toys. It is easier when you sort first, then count.

Worked Example 1

Coins: 1 quarter and 2 pennies

Step 1: Put in order.

Quarter, pennies

Step 2: Count.

Start with the quarter: \(25\)

Add 2 pennies: \(26, 27\)

Total: \(27\) cents

$$25 + 1 + 1 = 27$$

Worked Example 2

Coins: 1 dime, 1 nickel, and 3 pennies

Step 1: Put in order.

Dime, nickel, pennies

Step 2: Count.

Start with the dime: \(10\)

Add the nickel: \(15\)

Add 3 pennies: \(16, 17, 18\)

Total: \(18\) cents

$$10 + 5 + 1 + 1 + 1 = 18$$

Worked Example 3

Coins: 2 quarters, 1 dime, and 1 penny

Step 1: Put in order.

Quarter, quarter, dime, penny

Step 2: Count.

Start with 2 quarters: \(25, 50\)

Add 1 dime: \(60\)

Add 1 penny: \(61\)

Total: \(61\) cents

$$25 + 25 + 10 + 1 = 61$$

Worked Example 4

Coins: 1 quarter, 2 dimes, 1 nickel, and 4 pennies

Step 1: Put in order.

Quarter, dime, dime, nickel, pennies

Step 2: Count.

Start with the quarter: \(25\)

Add 1 dime: \(35\)

Add 1 more dime: \(45\)

Add 1 nickel: \(50\)

Add 4 pennies: \(51, 52, 53, 54\)

Total: \(54\) cents

$$25 + 10 + 10 + 5 + 1 + 1 + 1 + 1 = 54$$

Helpful counting tips

• Count quarters by 25s: \(25, 50, 75, 100\)
• Count dimes by 10s: \(10, 20, 30, 40\)
• Count nickels by 5s: \(5, 10, 15, 20\)
• Count pennies by 1s: \(1, 2, 3, 4\)

If you have more than one of the same coin, you can count those together first.

For example, 3 dimes are:

$$10 + 10 + 10 = 30$$

And 2 nickels are:

$$5 + 5 = 10$$

Then add the other coins after that.

Watch out for these mistakes:

• Do not count the number of coins. Count the value of the coins.
• A dime is worth 10 cents, even though it is smaller than a nickel.
• Put coins in order first so you do not forget any.
• Add pennies at the end because they are easiest to count by ones.

Try this thinking:

If you see a quarter, a nickel, and 2 pennies, say:

“Quarter is \(25\), nickel makes \(30\), 2 more pennies makes \(32\).”

So the total is \(32\) cents.

Another way to say the steps:

• Sort the coins.
• Start big.
• Count on.
• Check your total.

Let’s review with one more quick example.

Coins: 1 nickel, 1 quarter, 1 dime, 2 pennies

First, put them in order: quarter, dime, nickel, pennies.

Now count:

\(25\), \(35\), \(40\), \(41\), \(42\)

Total: \(42\) cents

Summary

To count mixed coin collections, first put the coins in order from highest value to lowest value: quarter, dime, nickel, penny.

Then count on using each coin’s value: quarters add \(25\), dimes add \(10\), nickels add \(5\), and pennies add \(1\).

This strategy helps you count money quickly, carefully, and correctly.

Equivalent Coin Combinations means making the same amount of money in different ways.

For example, you can make 10 cents with 1 dime, or with 2 nickels. These are equivalent coin combinations because they have the same value.

When we work with coins, we care about the value, not just how many coins there are. One coin can be worth more than many other coins.

Let’s remember the value of each coin:

• Penny = 1 cent
• Nickel = 5 cents
• Dime = 10 cents
• Quarter = 25 cents

If two groups of coins add to the same number of cents, then they are equivalent.

We can write that like this:

$$10\text{ cents} = 10\text{ cents}$$

Even if the coins look different, the total value can still be the same.

How to find equivalent coin combinations

1. Pick the total amount you want to make.
2. Add the value of one group of coins.
3. Make a different group of coins.
4. Check if both groups have the same total.

A good way to check is to count by coin values:

• Count pennies by 1s
• Count nickels by 5s
• Count dimes by 10s
• Count quarters by 25s

Important idea: Different coins can still make the same total.

Here are some simple equivalent amounts:

• 5 cents = 1 nickel = 5 pennies
• 10 cents = 1 dime = 2 nickels = 10 pennies
• 25 cents = 1 quarter = 2 dimes and 1 nickel = 5 nickels

Let’s look at some worked examples.

Example 1: Make 5 cents in two ways

First way: 1 nickel

$$5 = 5$$

Second way: 5 pennies

$$1+1+1+1+1=5$$

So, 1 nickel and 5 pennies are equivalent coin combinations.

Example 2: Make 10 cents in three ways

Way 1: 1 dime

$$10=10$$

Way 2: 2 nickels

$$5+5=10$$

Way 3: 1 nickel and 5 pennies

$$5+1+1+1+1+1=10$$

All three groups equal 10 cents. That means all three are equivalent coin combinations.

Example 3: Are these equivalent?

Group A: 1 dime and 1 nickel

Count the value:

$$10+5=15$$

Group B: 15 pennies

Count the value:

$$1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=15$$

Both groups are worth 15 cents.

So yes, they are equivalent.

Example 4: Make 25 cents in different ways

Way 1: 1 quarter

$$25=25$$

Way 2: 2 dimes and 1 nickel

$$10+10+5=25$$

Way 3: 5 nickels

$$5+5+5+5+5=25$$

Way 4: 25 pennies

$$1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=25$$

All of these are different coin combinations, but they all make 25 cents.

Tips for solving problems

• Start with the biggest coin you know.
• Count carefully to find the total value.
• Check your work by adding again.
• Remember: the same amount can be made with different coins.

Watch out for this mistake: Do not just count the number of coins.

For example, 2 coins are not always worth more than 1 coin.

1 dime is 10 cents, but 2 pennies are only 2 cents.

So we must count the value, not the number of coins.

Try thinking about these:

• Can you make 10 cents with a dime? Yes.
• Can you make 10 cents with nickels? Yes, 2 nickels.
• Can you make 10 cents with pennies? Yes, 10 pennies.

That shows one amount can have many equivalent coin combinations.

Summary

Equivalent coin combinations are different groups of coins with the same value. To find them, add the coin values and compare the totals. If the totals match, the combinations are equivalent.

Cent and Dollar Notation

Money amounts can be written in two different ways. We can write an amount in cents using the cent symbol 2, or we can write an amount in dollars and cents using the dollar sign $ and a decimal point.

Learning how to write money the right way helps us read prices, count money, and understand how much things cost.

1 dollar equals 100 cents.

We can write that like this:

$$1\text{ dollar} = 100\text{ cents}$$

That means 100 pennies make 1 dollar.

Ways to write money

• Cents only: use the cent symbol. Example: 452
• Dollars and cents: use the dollar sign and decimal point. Example: $0.45

These two amounts are the same:

$$45\text{ cents} = 45\text{2} = \$0.45$$

What does the decimal point mean?

When we write money with a dollar sign, the decimal point separates dollars from cents.

• The number before the decimal point tells the dollars.
• The number after the decimal point tells the cents.

In $3.25:

• 3 means 3 dollars
• 25 means 25 cents

So, $3.25 means 3 dollars and 25 cents.

Important rule: When writing dollars and cents, we write two digits after the decimal point.

For example:

• $0.05 means 5 cents
• $0.50 means 50 cents
• $2.07 means 2 dollars and 7 cents

Notice something important:

• 5 cents is $0.05, not $0.5
• 7 cents is $0.07, not $0.7

That is because money needs two digits for cents.

How to change cents into dollar notation

1. Think about how many dollars there are in the amount.
2. Write the dollar sign.
3. Put the dollars before the decimal point.
4. Put the cents after the decimal point using two digits.

If the amount is less than 1 dollar, write 0 before the decimal point.

For example:

• 8 cents = $0.08
• 32 cents = $0.32
• 100 cents = $1.00
• 125 cents = $1.25

How to change dollar notation into words or cents

Read the number before the decimal point as dollars. Read the two digits after the decimal point as cents.

For example:

• $4.12 = 4 dollars and 12 cents
• $0.30 = 30 cents
• $5.00 = 5 dollars

Worked Example 1

Write 9 cents in dollar notation.

9 cents is less than 1 dollar, so we write 0 dollars first.

Then we write 9 cents as 09.

$$9\text{ cents} = \$0.09$$

Worked Example 2

Write 68 cents in dollar notation.

68 cents is less than 1 dollar, so we write 0 before the decimal point.

Then we write 68 after the decimal point.

$$68\text{ cents} = \$0.68$$

Worked Example 3

Write 1 dollar and 7 cents in dollar notation.

We write the 1 before the decimal point because it is 1 dollar.

We write 7 cents as 07 after the decimal point.

$$1\text{ dollar and }7\text{ cents} = \$1.07$$

Worked Example 4

What amount is $2.45?

The 2 means 2 dollars.

The 45 means 45 cents.

So the amount is 2 dollars and 45 cents.

It is also:

$$\$2.45 = 2\text{ dollars and }45\text{ cents}$$

Tips to remember

• Use 2 when you write only cents.
• Use $ and a decimal point when you write dollars and cents together.
• Always write two digits after the decimal point.
• If there are no dollars, write 0 before the decimal point.

Lets look at some matching amounts

• 32 = $0.03
• 202 = $0.20
• 752 = $0.75
• 1002 = $1.00
• 1502 = $1.50

Brief Summary

Money can be written in cents or in dollars and cents. The cent symbol 2 is used for cents, and the dollar sign with a decimal point is used for dollars and cents. Remember that 100 cents equals 1 dollar, and always write two digits after the decimal point when using dollar notation.

Target Coin Combinations means choosing the exact coins needed to make a money amount.

When we buy something, we need enough money to pay for it. Sometimes we want to use exact change. Exact change means the coins add up to exactly the price—no more and no less.

In this lesson, we will practice making a target amount using coins. A target amount is the money amount we want to make, like 12 cents or 37 cents.

Know your coins first.

• Penny = \(1\) cent
• Nickel = \(5\) cents
• Dime = \(10\) cents
• Quarter = \(25\) cents

When we make a target amount, it helps to:

1. Look at the price.
2. Start with the biggest coin you can use.
3. Add more coins until you reach the target amount.
4. Check that your coins add to the exact total.

Start with bigger coins.

Using bigger coins first can make counting easier. For example, if you need to make 18 cents, you might start with a dime because 10 is a big part of 18.

Then ask: How many more cents do I need? This helps you choose the next coin.

Count on to the target.

Suppose you have 10 cents and need 18 cents. Count on:

\(10\), \(15\), \(16\), \(17\), \(18\)

That means you can add one nickel and three pennies.

More than one answer can be correct.

Sometimes a target amount can be made in different ways. For example, 10 cents can be made with:

• 1 dime
• 2 nickels
• 10 pennies
• 1 nickel and 5 pennies

If the coins add up to the target amount, the combination is correct.

Worked Example 1: Make 7 cents

We want exact change for 7 cents.

Start with the biggest coin you can use. A nickel is 5 cents. That is a good start.

Now find how many more cents are needed:

$$7 - 5 = 2$$

We need 2 more cents, so we add 2 pennies.

Answer: 1 nickel and 2 pennies

Check:

$$5 + 1 + 1 = 7$$

Worked Example 2: Make 14 cents

We want exact change for 14 cents.

Start with a dime because it is 10 cents.

Now find how many more cents are needed:

$$14 - 10 = 4$$

We need 4 more cents, so we add 4 pennies.

Answer: 1 dime and 4 pennies

Check:

$$10 + 1 + 1 + 1 + 1 = 14$$

There is another correct way too:

2 nickels and 4 pennies

Check:

$$5 + 5 + 1 + 1 + 1 + 1 = 14$$

Worked Example 3: Make 18 cents

We want exact change for 18 cents.

Start with a dime:

$$18 - 10 = 8$$

Now we need 8 more cents.

Use a nickel:

$$8 - 5 = 3$$

Now we need 3 more cents.

Add 3 pennies.

Answer: 1 dime, 1 nickel, and 3 pennies

Check:

$$10 + 5 + 1 + 1 + 1 = 18$$

Worked Example 4: Make 37 cents

We want exact change for 37 cents.

Start with the biggest coin you can use. A quarter is 25 cents.

Now find how many more cents are needed:

$$37 - 25 = 12$$

Now make 12 cents. A dime is 10 cents.

$$12 - 10 = 2$$

Now we need 2 more cents, so add 2 pennies.

Answer: 1 quarter, 1 dime, and 2 pennies

Check:

$$25 + 10 + 1 + 1 = 37$$

Helpful tips

• Say the coin values aloud. Quarter 25, dime 10, nickel 5, penny 1.
• Use the biggest coin first when it fits the amount.
• Stop at the exact amount. Do not go over the target.
• Check by adding. Make sure your coins match the price.

Let’s think about a real shopping problem.

A juice box costs 16 cents. What exact coins could you use?

Start with a dime:

$$16 - 10 = 6$$

Then use a nickel:

$$6 - 5 = 1$$

Then use 1 penny.

One correct answer: 1 dime, 1 nickel, and 1 penny

Check:

$$10 + 5 + 1 = 16$$

How to check your own work

1. Write or say the value of each coin.
2. Add the values together.
3. Ask: Does it equal the price?
4. If yes, you found an exact coin combination.

Summary

To make a target coin combination, choose coins that add up to the exact price. Start with bigger coins like quarters, dimes, and nickels, then use pennies if needed. Always check by adding all the coin values together.

Making Change from One Dollar

Sometimes an item costs less than one dollar, and you pay with a $1 bill. The money you get back is called change.

To make change from one dollar, we need to know that 1 dollar = 100 cents.

So if something costs 32 cents, you can think: 100 cents - 32 cents. The answer tells how much change you should get back.

What does change mean?

Change is the money you get back after you pay for something. If you pay with more money than the item costs, the cashier gives you back the extra amount.

When you pay with a $1 bill, you are paying with 100 cents. Your job is to find how many cents are left after paying for the item.

Two ways to find change

There are two good ways to solve these problems:

• Subtract from 100 cents
• Count up to 100 cents

Way 1: Subtract from 100 cents

If an item costs 45 cents, subtract:

So the change is 55 cents.

Way 2: Count up to 100 cents

If an item costs 45 cents, start at 45 and count up to 100.

1. 45 to 50 is 5 cents
2. 50 to 100 is 50 cents

Then add the jumps:

So the change is 55 cents.

Helpful idea: Make a friendly number first

When you count up, it often helps to jump to the next ten first.

For example, if something costs 68 cents:

• 68 to 70 is 2 cents
• 70 to 100 is 30 cents

Add the jumps:

So the change is 32 cents.

Worked Example 1

A pencil costs 23 cents. You pay with $1. How much change do you get?

Use subtraction:

You get 77 cents in change.

Worked Example 2

A sticker costs 58 cents. You pay with $1. How much change do you get?

Use counting up:

• 58 to 60 is 2 cents
• 60 to 100 is 40 cents

Add them:

You get 42 cents in change.

Worked Example 3

A juice box costs 76 cents. You pay with $1. How much change do you get?

Use subtraction:

You get 24 cents in change.

You can check by counting up:

• 76 to 80 is 4 cents
• 80 to 100 is 20 cents

Then:

Worked Example 4

A toy car costs 89 cents. You pay with $1. How much change do you get?

Count up to 100:

• 89 to 90 is 1 cent
• 90 to 100 is 10 cents

Add the jumps:

You get 11 cents in change.

Thinking about coins

After you find the change, you may also think about which coins could make that amount.

For example, 24 cents could be:

• 2 dimes and 4 pennies
• 1 dime, 2 nickels, and 4 pennies

There can be more than one way to make the change with coins.

Tips to remember

• $1 = 100 cents
• Change means the money you get back
• You can subtract the cost from 100
• You can count up from the cost to 100
• Jumping to the next ten can make counting easier

Let’s check our thinking

If an item costs only a little, your change will be a lot. If an item costs almost $1, your change will be a little.

For example:

• If something costs 12 cents, your change is large: $$100 - 12 = 88$$
• If something costs 95 cents, your change is small: $$100 - 95 = 5$$

This helps you see if your answer makes sense.

Summary

Making change from one dollar means finding how many cents are left from 100 cents. You can do this by subtracting the cost from 100 or by counting up from the cost to 100.

Both ways work. Choose the way that feels easiest to you, and always check if your answer makes sense.