Learn on PengienVision, Mathematics, Grade 4Chapter 12: Understand and Compare Decimals

Lesson 5: Solve Word Problems Involving Money

Property Money values can be written as decimals. Since 100 cents make up 1 dollar, any number of cents is that hundredth of a dollar. $$1 \text{ cent} = \frac{1}{100} \text{ of a dollar} = \$0.01$$.

Section 1

Writing Money Values as Decimals

Property

Money values can be written as decimals. Since 100 cents make up 1 dollar, any number of cents is that hundredth of a dollar.

1 cent=1100 of a dollar=$0.011 \text{ cent} = \frac{1}{100} \text{ of a dollar} = \$0.01

Examples

  • 1 quarter is 25 cents, which is written as $0.25\$0.25.
  • 3 dimes and 4 pennies is 30+4=3430 + 4 = 34 cents, which is written as $0.34\$0.34.
  • 2 dollars, 1 quarter, and 1 nickel is $2.00+$0.25+$0.05\$2.00 + \$0.25 + \$0.05, which is written as $2.30\$2.30.

Explanation

To write a money amount in decimal form, we use a decimal point to separate the dollars from the cents. The numbers to the right of the decimal point represent the cents, which are parts of a whole dollar. Since there are 100 cents in a dollar, we always use two decimal places for cents. For example, 50 cents is half of a dollar, written as $0.50\$0.50.

Section 2

Adding and Subtracting Money by Denomination

Property

To find the total value or the difference between two sets of money, group the bills and coins by their denomination. Calculate the total value for each denomination separately, then sum these values for the final amount. For subtraction, you may need to make change from a larger denomination.

Examples

  • Addition: You have two $10\$10 bills, one $5\$5 bill, and 3 quarters. Your friend has one $10\$10 bill, two $5\$5 bills, and 2 quarters. To find the total, combine like denominations:
(2+1)×$10.00+(1+2)×$5.00+(3+2)×$0.25 (2+1) \times \$10.00 + (1+2) \times \$5.00 + (3+2) \times \$0.25
3×$10.00+3×$5.00+5×$0.25=$30.00+$15.00+$0.25+$0.25+$0.25+$0.25+$0.25=$46.25 3 \times \$10.00 + 3 \times \$5.00 + 5 \times \$0.25 = \$30.00 + \$15.00 + \$0.25 + \$0.25+ \$0.25+ \$0.25+ \$0.25 = \$46.25
  • Subtraction: You have one $20\$20 bill and want to buy an item that costs $12.50\$12.50. You can think of the $20\$20 bill as one $10\$10 bill, one $5\$5 bill, and five $1\$1 bills (or other combinations). To find the change from $12.50\$12.50 ($10+$2+2\$10 + \$2 + 2 quarters):
$20.00−$12.50=($10+$5+$1+$1+$1+$1+$1)−($10+$2+$0.25+$0.25)=$7.50\$20.00 - \$12.50 = (\$10 + \$5 + \$1 + \$1 + \$1 + \$1 + \$1) - (\$10 + \$2 + \$0.25 + \$0.25) = \$7.50

Your change could be one $5\$5 bill, two $1\$1 bills, and 2 quarters.

Explanation

This method mirrors how we handle physical cash by grouping similar bills and coins together. First, count the number of each denomination (e.g., all the ten-dollar bills, all the quarters). Then, find the total value for each group and add these values together to get the grand total. This strategy helps reinforce the value of each bill and coin and provides a practical way to perform calculations without solely relying on decimal algorithms.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Writing Money Values as Decimals

Property

Money values can be written as decimals. Since 100 cents make up 1 dollar, any number of cents is that hundredth of a dollar.

1 cent=1100 of a dollar=$0.011 \text{ cent} = \frac{1}{100} \text{ of a dollar} = \$0.01

Examples

  • 1 quarter is 25 cents, which is written as $0.25\$0.25.
  • 3 dimes and 4 pennies is 30+4=3430 + 4 = 34 cents, which is written as $0.34\$0.34.
  • 2 dollars, 1 quarter, and 1 nickel is $2.00+$0.25+$0.05\$2.00 + \$0.25 + \$0.05, which is written as $2.30\$2.30.

Explanation

To write a money amount in decimal form, we use a decimal point to separate the dollars from the cents. The numbers to the right of the decimal point represent the cents, which are parts of a whole dollar. Since there are 100 cents in a dollar, we always use two decimal places for cents. For example, 50 cents is half of a dollar, written as $0.50\$0.50.

Section 2

Adding and Subtracting Money by Denomination

Property

To find the total value or the difference between two sets of money, group the bills and coins by their denomination. Calculate the total value for each denomination separately, then sum these values for the final amount. For subtraction, you may need to make change from a larger denomination.

Examples

  • Addition: You have two $10\$10 bills, one $5\$5 bill, and 3 quarters. Your friend has one $10\$10 bill, two $5\$5 bills, and 2 quarters. To find the total, combine like denominations:
(2+1)×$10.00+(1+2)×$5.00+(3+2)×$0.25 (2+1) \times \$10.00 + (1+2) \times \$5.00 + (3+2) \times \$0.25
3×$10.00+3×$5.00+5×$0.25=$30.00+$15.00+$0.25+$0.25+$0.25+$0.25+$0.25=$46.25 3 \times \$10.00 + 3 \times \$5.00 + 5 \times \$0.25 = \$30.00 + \$15.00 + \$0.25 + \$0.25+ \$0.25+ \$0.25+ \$0.25 = \$46.25
  • Subtraction: You have one $20\$20 bill and want to buy an item that costs $12.50\$12.50. You can think of the $20\$20 bill as one $10\$10 bill, one $5\$5 bill, and five $1\$1 bills (or other combinations). To find the change from $12.50\$12.50 ($10+$2+2\$10 + \$2 + 2 quarters):
$20.00−$12.50=($10+$5+$1+$1+$1+$1+$1)−($10+$2+$0.25+$0.25)=$7.50\$20.00 - \$12.50 = (\$10 + \$5 + \$1 + \$1 + \$1 + \$1 + \$1) - (\$10 + \$2 + \$0.25 + \$0.25) = \$7.50

Your change could be one $5\$5 bill, two $1\$1 bills, and 2 quarters.

Explanation

This method mirrors how we handle physical cash by grouping similar bills and coins together. First, count the number of each denomination (e.g., all the ten-dollar bills, all the quarters). Then, find the total value for each group and add these values together to get the grand total. This strategy helps reinforce the value of each bill and coin and provides a practical way to perform calculations without solely relying on decimal algorithms.