 # 2.08 Practical problems with decimal operations

Lesson

Once we are comfortable performing operations with decimals, we can think about how to manipulate decimal quantities that we come across in the real world. Exchanging money, measuring lengths and weights, and recording times are all areas that make use of decimal numbers.

#### Worked example

##### question 1

Lucy is a transport truck driver and pulls into a gas station that advertises unleaded gas for $134.8$134.8 cents per gallon. She fills up her truck with $41.57$41.57 gallons of gas and picks up a $\$5.95$$5.95 bottle of engine oil before paying the cashier. How much will Lucy have to pay? Give your answer in dollars, to the nearest cent. Think: We know the cost of gas in cents per gallon, and the amount of gas in gallons that Lucy gets. We also know the cost in dollars of the engine oil. Our goal is to combine these quantities to get the total cost in dollars. Do: A single gallon of gas costs 134.8134.8 cents, which is \frac{134.8}{100}=\1.348134.8100=1.348. So the cost in dollars for 41.5741.57 gallons will be given by the product 1.348\times41.571.348×41.57. Next, we can add the cost of the engine oil, and round the total to two decimal places. The working out for this calculation is shown below.  \text{Total cost }Total cost == \text{gas price }\times\text{gas purchased }+\text{engine oil price }gas price ×gas purchased +engine oil price == \frac{134.8}{100}\times41.57+5.95134.8100​×41.57+5.95 Substitute the given information == 1.348\times41.57+5.951.348×41.57+5.95 Simplify the fraction == 56.03636+5.9556.03636+5.95 Evaluate the multiplication == 61.9863661.98636 Evaluate the addition == 61.9961.99 Round to the nearest cent The total cost for the gas and the engine oil is \61.98636$$61.98636, which is $\$61.99$$61.99 when rounded to the nearest cent. Reflect: Instead of converting the price of gas to dollars per gallon, we could have found the total cost in cents and converted to dollars at the end. Since we are working in cents rather than dollars, we will round to the nearest whole number rather than to two decimal places. The working out for this method is shown below.  \text{Total cost }Total cost == \text{gas price }\times\text{gas purchased }+\text{engine oil price }gas price ×gas purchased +engine oil price == 134.8\times41.57+5.95\times100134.8×41.57+5.95×100 Both quantities are in cents == 5603.636+5955603.636+595 Evaluate each multiplication separately == 6198.6366198.636 Evaluate the addition == 61996199 Round to the nearest whole number Our total cost is 61996199 cents, which is the same as \61.99$$61.99, as expected.

Strategies for solving real world problems

The solution to many real world problems will eventually involve some kind of calculation, but there is a lot we can do before and after this calculation that can make us more confident our answer is correct.

• What are the quantities that we are combining?
• What units do we expect the answer to have?
• What operations will combine the relevant quantities to produce the expected units?
• What do we estimate that a reasonable answer will be?
• Does the answer we calculate seem appropriate in the context?

#### Practice questions

##### Question 2

Harry buys an item from the school canteen for $\$3.203.20. If he pays for it with a five dollar note, how much change will he get back?

##### Question 3

How many $0.38$0.38 L bottles can be filled from a barrel which holds $41.8$41.8 L?

##### Question 4

At midnight in Alaska, the temperature in Juneau is $36.8$36.8 degrees Fahrenheit.
Each hour after that the temperature decreases by $2.43$2.43 degrees until the sun comes up.
What is the temperature $2$2 hours after midnight?

### Outcomes

#### 6.NS.3

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.