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Decomposing Money

Introduction

Money, which includes notes and coins, is the means of exchange that is used to exchange goods and services. We all deal with money on a daily basis in one way or another, and in order to reliably handle that money, we often turn on our mathematical abilities.

Each government has its currency, and the word “money” is frequently used to refer to payment for goods in the form of coins or paper currency. For the purchase of products or the use of services, money is exchanged between different parties. We are frequently required to render the precise amount to be paid or exchanged. We need to be aware of a currency’s denominations to achieve that.

Let us say, for instance, that you have a \$350 bill to pay. You have some \$100, \$50, and \$20 bills. You must do money decomposition to ensure that you pay the exact amount because no currency is exactly \$350. You can therefore give the clerk three notes totaling \$100 and 1 note totaling \$50. The \$20 banknotes might be used in other combinations as well. This is only one illustration of how the decomposition of money works.

This article will define deconstructing money, discuss its significance, and walk through some related exercises.

What is Decomposing Money?

Definition

Money is said to be decomposed when it is expressed as a collection of smaller denominations in any currency. Decomposing money is the process of separating or splitting a large quantity of money into several small units of money. It is important to understand the many denominations that each currency has to provide in order to break down money.

For example, there are seven denominations in the American paper currency: \$1, \$2, \$5, \$10, \$20, \$50, and \$100. So, if you must pay \$15, you may pay two notes of \$10 and \$5. Another option is to have 1 note of \$10, 2 notes of \$2 and 1 note of \$1. Depending on the available money you have, you may decompose the money needed for the transaction.

Cents, which are available in coins, are also used in addition to dollar bills. Penny (1¢ ), nickel ( 5¢ ), dime (10¢ ), and quarters ( 25 ¢ ) are the denominations available. The following breakdown may be taken into account if you wish to make \$1: 100 pennies, 20 nickels, 10 dimes, or 4 quarters.

For instance, if you want to purchase a sweet that costs 75 cents, you must give the vendor three pieces of 25 cents. There may be various ways to pay the seller the exact amount if you have additional coins available.

How Do You Decompose Money?

Let us assume you are in the grocery store and have to pay \$725 for the items in your cart. If you have nine \$100 banknotes in your wallet, you must give the shopkeeper eight of them in exchange for three \$50, \$20, and \$5 notes. This is only one instance of how decomposing money works.

The breakdown of various sums of money is shown in the table below, taking the following notes into account: \$1, \$2, \$5, \$10, \$20, \$50, and \$100.

\$1350= ( \$100×13 ) +( \$50×1 )
\$1300= ( \$100×13 )
\$1275= ( \$100×12 ) + ( \$50×1 ) + ( \$20×1 ) +( \$5×1 )
\$1150= ( \$100×11 ) + ( \$50×1 )
\$914= ( \$100×9 ) + ( \$10×1 ) + ( \$2×2 )
\$713= ( \$100×7 ) + ( \$10×1 ) + ( \$2×1 ) +( \$1×1 )
\$541= ( \$100×5 ) + ( \$20×2 ) + ( \$1×1 )
\$485= ( \$100×4 ) + ( \$10×8 ) + ( \$5×1 )
\$352= ( \$100×3 ) + ( \$50×1 ) + ( \$1×2 )
\$247= ( \$100×2 ) + ( \$20×2 ) + ( \$5×1 ) + ( \$2×2 )
\$158= ( \$100×1 ) + ( \$50×1 ) + ( \$5×1 ) +( \$1×3 )
\$99= ( \$50×1 ) + ( \$20×2 ) + ( \$5×1 ) +( \$2×2 )
\$63= ( \$50×1 ) + ( \$10×1 ) + ( \$2×1 ) +( \$1×1 )
\$52= ( \$50×1 ) + ( \$2×1 )
\$49= ( \$10×4 ) + ( \$5×1 ) + ( \$2×2 )
\$42= ( \$20×2 ) + ( \$1×2 )
\$24= ( \$10×1 ) + ( \$5×2 ) +(\$1×4 )

Importance of Decomposing Money

The ability to manage money is one of the essential life skills. The following are only a few of the many advantages of money decomposition.

( 1 ) Numerous fundamental ideas, such as place values and number counting, considerably assist us in managing the decomposition of money. Because purchasing transactions require the breakdown of money, it is vital to apply fundamental knowledge of the place value system and number counting to ensure the transaction is completed correctly. 

( 2 ) The idea of the decomposing of money is vital, from making a tiny purchase to putting large quantities of money in a bank account. 

( 3 ) By keeping track of how much money was spent and how much is still available, decomposing money is a skill that enhances money management. As a result, it enables us to manage our finances better, whether for personal budgeting or even for all business transactions. 

( 4 ) We put a particular amount of money in a bank each time we invest. It is essential to know how to divide or split up the money in these situations and what portion should be deposited. For instance, you have \$4000 and have decided to deposit \$1500 and invest \$2500. Decomposing your money is essential in this situation to make sure you separate your cash into that which is for investing and that which is for depositing.

Examples

Example 1

Add the money below and find the total amount.

( a )

( b )

( c )

( d )

( e )

( f )

Solution:

Let us show the breakdown of the sums of money.

( a ) ( \$100×3 ) + ( \$10×2 ) + ( \$5×1 )

=\$300+\$20+\$5

=\$235

( b ) ( \$100×1 ) + ( \$50×1 ) + ( \$20×1 ) + ( \$10×1 ) + ( \$5×1 )

=\$100+\$50+\$20+\$10+\$5

=\$185

( c ) ( \$50×1 ) + ( \$20×1 ) + ( \$10×1 ) + ( \$5×1 ) + ( \$5×1 ) + (\$2×1 )

=\$50+\$20+\$10+\$10+\$2

=\$92

( d ) ( \$100×2 ) + ( \$50×1 ) + ( \$20×2 ) + ( \$10×1 ) + ( \$5×2 )

=\$200+\$50+\$40+\$10+\$10

=\$310

( e ) ( \$100×1 ) + ( \$50×2 ) + ( \$20×2 ) + ( \$10×1 ) + ( \$5×3 ) + ( \$2×1 ) + (\$1 ×1 )

=\$100+\$100+\$40+\$10+\$15+\$2+\$1

=\$268

( f ) ( \$100×2 ) + ( \$50×2 ) + ( \$20×2 ) + ( \$10×2 ) + ( \$5×2 ) + ( \$2×1 ) + (\$1 ×1 )

=\$200+\$100+\$40+\$20+\$10+\$2+\$1

=\$373

Example 2

Write down how many dollars, in any combination of \$100, \$50, or \$20, are required to equal \$700.

Solution:

These are some of the possible money decompositions to get a total of \$700.

( a ) ( \$100×7 ) =\$700

( b ) ( \$100×6 ) + ( \$50×2 ) =\$700

( c ) ( \$100×5 ) + ( \$50×4 ) =\$700

( d ) ( \$100×6 ) + ( \$20×5 ) =\$700

( e ) ( \$100×5 ) + ( \$50×2 ) + ( \$20×5 ) =\$700

( f ) ( \$50 10 ) + (\$20 10 )  = \$700

( g ) ( \$100 ×4  ) + ( \$ 50 4 ) + ( \$20 5 ) = \$700

Example 3

Using any combinations of \$100, \$50, \$20, or \$10, identify how many further dollars are needed to reach the amount of \$450.

Solution:

These are a few alternative ways to break the \$450 into smaller amounts.

( a ) ( \$100×4 ) + ( \$50×1 ) =\$450

( b ) ( \$100×4 ) + ( \$20×2 ) + ( \$10×1 ) =\$450

( c ) ( \$100×3 ) + ( \$50×2 ) + (\$20×2 +( \$10×1 )=\$450

( d ) ( \$100×2 ) + ( \$50×4 ) + ( \$50×1 ) =\$450

( e ) ( \$100×4 ) + ( \$10×5 ) =\$450

( f ) ( \$50×8 ) + ( \$20×2 ) + ( \$10×1 ) =\$450

( g ) ( \$50×6 ) + ( \$20×5 ) + ( \$10×5 ) =\$450

Example 4

Fill in the blanks with the correct denomination so that the total value is the same as the given.

Use the denominations: \$1, \$2, \$5, \$10, \$20, \$50, and \$100.

( a )

( ____ 1 ) + ( ____ 2 ) + ( ____ ×1 ) = \$100

( b )

( ___ 2 ) + ( ____ 2 ) = \$50

( c )

( ____ × 3 ) + ( ____ 2 ) + ( ____ 1 ) = \$20

( d )

( ____ ×1 ) + ( ____ 2 ) + ( ____ ×1 ) = \$10

( e )

( ____ ×3 ) + ( ____ 2 ) + ( ____ ×4 ) = \$100

Solution:

The following solutions are available because we need to consider the denominations of \$1, \$2, \$5, \$10, \$20, \$50, and \$100. 

( a ) ( \$50 x 1 ) + ( \$20 x 2 ) + ( \$10 ×1 ) = \$100

( b ) ( \$20 x 2 ) + ( \$5 x 2 ) = \$50

( c ) ( \$5 × 3 ) + ( \$2 x 2 ) + ( \$1 x 1 ) = \$20

( d ) ( \$5 × 1 ) + ( \$2 x 2 ) + ( \$1 × 1 ) = \$10

( e ) ( \$20 × 3 ) + ( \$10 x 2 ) + ( \$5 × 4 ) = \$100

Example 5

In the table below, fill out Column 3 (How much change is left?). You can see your available funds in Column 1 and the cost of the item you want to purchase in Column 2. Use the denominations: \$1, \$2, \$5, \$10, \$20, \$50, and \$100 to show the possible combinations of money.

Available MoneyCost of the itemHow much change is left?

\$12

\$45

\$85

\$128

\$175

Solution:

We must deduct the values in order to find the change that remains in each item with the specified available fund. For instance, if the amount of money available is \$50 and the item to be purchased is \$38, the change must be \$50-\$38=\$12. The possible currency combination would be 1 note of \$10 and 1 note of \$2 since the change is \$12.

The complete table is shown below.

Available MoneyCost of the itemHow much change is left?
\$12\$20 – \$12 = \$8
\$5 + \$2 + \$1 = \$8
\$38\$50 – \$38 = \$12
\$10 + \$2 = \$12
\$85\$100 – \$85 = \$15
\$10 + \$5 = \$15

\$127
\$ 150 – \$127 = \$23
\$20 + \$2 + \$1 = \$23  

\$164
\$200 – \$164 = \$36
\$20 + \$10 + \$5 + \$1 = \$36

Example 6

Find two ways to pay the item’s stated price. Use the denominations: \$1, \$2, \$5, \$10, \$20, \$50, and \$100 to show the possible combinations of money.

( a ) French Bread \$3

( b ) Cereal \$4

( c ) Butter \$6

( d ) T-Bone Steak \$9

( e ) Fruits \$7

( f ) Salmon \$23

( g ) Clothes \$129

Solution:

Using the denominations, \$1, \$2, \$5, \$10, \$20, \$50, and \$100, the possible combinations of money in each item are listed below. Remember that there can be other possible combinations in addition to the ones listed below.

( a ) French Bread \$3

First Way: \$1 + \$1 + \$1 = \$3

Second Way: \$2 + \$1 = \$3

( b ) Cereal \$4

First Way: \$1 + \$1 + \$1 + \$1= \$4

Second Way: \$2 + \$2 = \$4

( c ) Butter \$6

First Way: \$2 + \$2 + \$1 + \$1 = \$6

Second Way: \$2 + \$1 + \$1 + \$1 + \$1 = \$6

( d ) T-Bone Steak \$9

First Way: \$5 + \$2 + \$2 = \$9

Second Way: \$5 + \$2 + \$1 + \$1 = \$9

( e ) Fruits \$7

First Way: \$5 + \$2 = \$7

Second Way: \$5 + \$1 + \$1 = \$7

( f ) Salmon \$23

First Way: \$20 + \$2 + \$1 = \$23

Second Way: \$10 + \$10 + \$2 + \$1 = \$23

( g ) Clothes \$129

First Way: \$100 + \$20 + \$5 + \$2 + \$2 = \$129

Second Way: \$100 + \$10 + \$10+ \$5 + \$2 + \$1 + \$1 = \$129

Example 7

After pooling their money, Amy, Bernadette, and Penny had 18 \$1 notes in total. If everyone gave at least \$2 and in a different quantity, calculate the five possible starting amounts they may have had.

Amy’s
Contribution
Bernadette’s
Contribution
Penny’s
Contribution
Total Contribution
\$18
\$18
\$18
\$18
\$18

Solution:

Given that Amy, Bernadette, and Penny each contributed at least \$2, let’s split the \$18 contribution among them. The five possible combinations are shown in the table below. These are just some of the many possible combinations.

Amy’s
Contribution
Bernadette’s
Contribution
Penny’s
Contribution
Total Contribution
( 1 )       \$10\$5\$3\$18
( 2 )         \$5\$5\$8\$18
( 3 )         \$9\$4\$5\$18
( 4 )         \$3\$7\$8\$18
( 5 )         \$6\$8\$4\$18

You will see that each of them gave more than \$2. There might be additional money combinations besides those in the table that add up to \$18. In this case, it matters that each of them contributed a minimum of \$2 and the required total of \$18.

Summary

Definition

Decomposing money is the process of separating or splitting a large quantity of money into several small units of money.

There are seven denominations in the American paper currency:  \$1, \$2, \$5, \$10, \$20, \$50, and \$100. 

Cents, which are available in coins, are also used in addition to dollar bills. Penny (1¢ ),  nickel ( 5¢ ), dime (10¢ ), and quarters ( 25 ¢ ) are the denominations available. The following breakdown may be taken into account if you wish to make \$1: 100 pennies, 20 nickels, 10 dimes, or 4 quarters.

Importance of Decomposing Money

The ability to manage money is one of the most important life skills. The following are only a few of the many advantages of money decomposition.

(1) It is important to break down cash while making purchases.

(2) Money must be broken down before being deposited into bank accounts.

(3) The ability to break down money is a skill that enhances money management.

(4) Investments require money to be broken down.

Frequently Asked Questions on Decomposing Money (FAQs)

What is meant by decomposing money? 

Decomposing money is the process of separating or splitting a large quantity of money into several small units of money. Money is said to be decomposed when it is expressed as a collection of smaller denominations in any currency. It is important to understand the many denominations that each currency has to provide in order to break down money. 

For instance, American paper money comes in seven different denominations: \$1, \$2, \$5, \$10, \$20, \$50, and \$100. Therefore, if you need to pay \$15, you can do so with two notes worth \$10 and \$5. Another choice is to have one \$10 note, two \$2 notes, and one \$1 note. You may divide up the money required for the transaction based on available funds.

Why is it essential to learn about decomposing money?

One of the most important life skills is having good money management abilities. The benefits of money decomposition are numerous; just a few are explained below.

The idea of decomposing money is vital, from making a tiny purchase to putting large quantities of money in a bank account. Because purchasing transactions require the breakdown of money, it is essential to apply fundamental knowledge of the place value system and number counting to ensure the transaction is completed correctly.  

Decomposing money is a skill that enhances money management by keeping track of how much was spent and how much is still available. As a result, it enables us to manage our finances better, whether for personal budgeting or even for all business transactions. Decomposing your money is essential in different situations to make sure you separate your cash into that which is for investing and that which is for depositing.

In which cases can I use decomposing money?

These are only a few of the various uses for decomposing money.

( 1 ) Making purchases shows how money is decomposing. When you need to pay for the products in your basket at the grocery shop, for instance, you may choose to give the shopkeeper the cash based on your available funds. You must pay the storekeeper the total amount in different possible money combinations. In a similar manner, the cashier has the option to use decomposition when issuing changes.

( 2 ) Money decomposition is also noticeable when depositing money into a bank. Even the deposit slip has a breakdown of the denominations when making a deposit transaction.

( 3 ) By tracking how much money was spent and how much money is still available, money decomposition is used in money management. Decomposing money is vital since managing finances is a key component of any household.

( 4 ) Money is an essential component in making investments. Any investor who wants to allocate their funds wisely in investment-related activities must understand how to decompose money properly.

How do you decompose \$100?

There are several ways to split \$100, depending on the sum of money available or the number of bills that must be paid

The listed below are just few examples of how \$100 could be divided into different denominations ( \$1, \$2, \$5, \$10, \$20, and \$50 ).

( a ) ( \$50×2 )=\$100

( b ) ( \$50×1 ) + ( \$20×2 ) + ( \$10×1 ) =\$100

( c ) ( \$20×5 ) =\$100

( d ) ( \$20×4 ) + ( \$5×4 ) =\$100

( e ) ( \$50×1 ) + ( \$10×5 ) =\$100

( f ) ( \$10×10 ) =\$100

( g ) ( \$10×6 ) + \$20×2 =\$100

What is the difference between money and currency?

Governments often issue money in the form of coins or paper currency, which is used to pay for commodities. Economists commonly understand money to be a frequently used form of exchange. A currency is a monetary system utilized in a nation. Alternatively, we might say that a currency is a monetary system used frequently, especially by citizens of that nation. Because numerous types of currencies are utilized in various countries, each one has its own currency value. Examples of currencies include US dollars (US\$), Euros (€), Japanese Yen (¥), Philippine Peso (₱), and Pounds Sterling (£).

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