# 2.04 Practical problems with rational numbers

Lesson

We have now see our four main operations, addition, subtraction, multiplication and division, with different types of rational numbers. What applications or practical problems can we solve using these skills? Before we jump into worded problems, here are reminders of what we have seen so far:

Operations with rational numbers
• Operations with integers can be done by considering the number line, using integer chips or using the rules about adjacent signs
• Operations with decimals require the decimal point to be lined up for addition and subtraction and place values to be considered for multiplication and division
• Operations with fractions require a common denominator for addition and subtraction, but not for multiplication and division - for both it can helpful to convert from mixed numbers to improper fractions

When working with practical problems, it is important to be able to identify keywords. Here is a list of a selection, what other ones can you think of?

Keywords

Addition: add, both, combined, increase, join, more, plus, sum, together, total

Subtraction: decrease, difference, fewer, left over, less, minus, subtract, take away

Multiplication: by, double, groups of, multiply, of, per, product, times

Division: divide, evenly, half, quotient, shared, split

There are many different strategies we can use to solve practical problems, we will just show one approach, but that doesn't mean there aren't other great ways!

#### Worked examples

##### Question 1

In the morning it was $5$5 $^\circ$°C and it went up $6$6 $^\circ$°C during the day and then went down $13$13 $^\circ$°C in the evening. Write an number sentence for the temperature throughout the day and evaluate it to find the temperature at the end of the day.

Think: We can represent this with integers as they are positive and negatives of whole numbers. Going "up" will be positive and going "down" will be negative.

Do:

 $5+6-13$5+6−13 Starting at $5$5 and then going up $6$6 and then down $13$13 $=$= $11-13$11−13 We can work from left to right $5+6=11$5+6=11 $=$= $-2$−2 We are subtracting a larger number from a smaller number, so expect a negative

The temperature at the end of the day was $-2$2 $^\circ$°C.

Reflect: How could you use a number line to draw out this scenario?

##### Question 2

How many $0.26$0.26 L glasses can a $20.8$20.8 L water bottle fill?

Think: We can are splitting the $20.8$20.8 L into a number of $0.26$0.26 L glasses. "Split" is one of our keywords which tells us we need to divide.

Do: Write this as a division problem:

 $20.8\div0.26$20.8÷​0.26 We are dividing the $20.8$20.8 L among the $0.26$0.26 L glasses $=$= $2080\div26$2080÷​26 We can multiply both by $100$100 to eliminate the decimal $=$= $80$80 Use long division to evaluate

$80$80 glasses can be filled.

Reflect: If we were only asked to estimate, what might we estimate? Is our answer reasonable?

##### Question 3

A baker has $2$2 cups of flour left. A cake requires $1\frac{1}{5}$115 cups of flour and cookies require $\frac{3}{4}$34 cups of flour. Will the baker have enough flour? How much will they have left or be short? Write a number sentence to justify your answer.

Think: If we start with $2$2 cups and use $1\frac{1}{5}$115 cups and then another $\frac{3}{4}$34 of a cup, we need to subtract $1\frac{1}{5}$115 and $\frac{3}{4}$34 from $2$2.

Do:

 $2-1\frac{1}{5}-\frac{3}{4}$2−115​−34​ Taking away the two different amount of flour $=$= $2-\frac{6}{5}-\frac{3}{4}$2−65​−34​ Write $1\frac{1}{5}$115​ as an improper fraction $=$= $\frac{40}{20}-\frac{24}{20}-\frac{15}{20}$4020​−2420​−1520​ Write with a common denominator $=$= $\frac{16}{20}-\frac{15}{20}$1620​−1520​ Subtract from left to right $=$= $\frac{1}{20}$120​ Subtract from left to right

We get positive $\frac{1}{20}$120, so the baker will have enough for both the cake and the cookies with $\frac{1}{20}$120 of a cup to spare.

Reflect: How could you solve this problem using another strategy?

#### Practice questions

##### Question 4

Consider the following phrase:

The quotient of $-3$3 and the sum of $7$7 and $6$6 .

1. Without simplifying the result, translate this sentence into a mathematical expression.

2. Evaluate the expression.

##### Question 5

Jenny takes out a loan of $\$2200$$2200. She pays back \42.60$$42.60 each month and doesn't have to pay interest.

1. If she has made $5$5 repayments so far, how much does Jenny still owe?

##### Question 6

A bottle is $\frac{2}{7}$27 full of cordial. If $230$230 milliliters of cordial is added to it, the bottle is $\frac{5}{6}$56 full. How many milliliters does the bottle hold when full?

### Outcomes

#### NY-7.NS.3

Solve real-world and mathematical problems involving the four operations with rational numbers. Notes: Computations with rational numbers extend the rules for manipulating fractions to complex fractions limited to (a/b)/(c/d) where a, b, c, and d are integers and b, c, and d ≠ 0.

#### NY-7.EE.3

Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies.