The numbers involved in solving real-world problems are not always nice whole numbers or even integers. We need a variety of numbers to be able to accurately represent real-world situations. That is where rational numbers come in.

Rational numbers

The set of all numbers that can be written as the ratio of two integers with a non-zero denominator

Example:

5\dfrac{2}{3}, \dfrac{15}{4}, -15, 0, 0.5

\sqrt{25}, 72, 81 \%, 0. \overline{22}

When working with real-world problems, it's important to be able to identify keywords. These keywords can help us determine which operation(s) can be used to solve the problem. Here are some examples of keywords which indicate the use of a certain operation.

addition	subtraction	multiplication	division
more	less	product	equally shared
add	subtract	by	in each
all together	how many left	times	per
total	difference	groups of	divided by
increase	decrease	double	half
combined	fewer	multiply	split

Estimation can be a very useful strategy when solving real-world problems. They can help us quickly find a solution that is close to the answer and depending on the context, we may not need an exact answer, an approximation may be good enough.

Examples

Example 1

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

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

Worked Solution

Create a strategy

Subtract the total repaid from the total amount borrowed.

Apply the idea

\displaystyle \text{Balance}	\displaystyle =	\displaystyle 2200-(42.60\cdot 5)	Subtract the total repaid from borrowed amount
	\displaystyle =	\displaystyle 2200-213	Evaluate the multiplication
	\displaystyle =	\displaystyle \$1987.00	Evaluate

Example 2

A bottle is \dfrac27 full of orange juice. If 230 milliliters of orange juice is added to it, the bottle is \dfrac56 full. How many milliliters does the bottle hold when full?

Worked Solution

Create a strategy

Divide the added amount by the difference of first full amount from the second full amount.

Apply the idea

\displaystyle \text{Full bottle}	\displaystyle =	\displaystyle 230\div\left({\dfrac56-\dfrac27}\right)	Divide the added amount by the amount difference
	\displaystyle =	\displaystyle 230\div\left(\dfrac{35}{42}-\dfrac{12}{42}\right)	Find the common denominator
	\displaystyle =	\displaystyle 230\div\dfrac{23}{42}	Evaluate the parenthesis
	\displaystyle =	\displaystyle \dfrac{230\cdot 42}{1\cdot23}	Multiply by the reciprocal
	\displaystyle =	\displaystyle \dfrac{9660}{23}	Evaluate the multiplication
	\displaystyle =	\displaystyle 420\text{ mL}	Evaluate

Reflect and check

When checking our answer, we can use estimation to verify if our answer seems reasonable. For example, we can say that \dfrac{5}{6} is not much different from \dfrac{5}{7}, which allows us to find a common denominator.

\displaystyle \text{Full bottle}	\displaystyle \approx	\displaystyle 230\div\left({\dfrac57-\dfrac27}\right)	Divide the added amount by an estimated difference
	\displaystyle =	\displaystyle 230\div\dfrac{3}{7}	Evaluate the subtraction

After subtracting, we get \dfrac{3}{7}, which is close to \dfrac{3}{6} or \dfrac{1}{2}.

\displaystyle 230\div\dfrac{3}{7}	\displaystyle \approx	\displaystyle 230\div\dfrac{1}{2}	Estimate \dfrac{3}{7} as \dfrac{1}{2}
	\displaystyle =	\displaystyle 230\cdot 2	Multiply by the reciprocal
	\displaystyle =	\displaystyle 460 \text{ mL}	Evaluate the multiplication

This estimation is close to our actual answer of 420 \text{ mL}, confirming that our answer is reasonable.

It is important to keep in mind that, while estimation is a good strategy for verifying our answer, it is not an ideal strategy for actually finding the capacity of the bottle. In this case, we slightly overestimated the result, which might lead to overfilling the bottle if we were to rely solely on the estimation. Therefore, it's important to use the precise calculation we performed earlier to determine the exact capacity of the bottle and ensure that we don't overfill it.

Idea summary

Recall the following operations and keywords when solving real-world problems.

addition	subtraction	multiplication	division
more	less	product	equally shared
add	subtract	by	in each
all together	how many left	times	per
total	difference	groups of	divided by
increase	decrease	double	half
combined	fewer	multiply	split

Outcomes

7.CE.1

The student will estimate, solve, and justify solutions to multistep contextual problems involving operations with rational numbers.

7.CE.1a

Estimate, solve, and justify solutions to contextual problems involving addition, subtraction, multiplication, and division with rational numbers expressed as integers, fractions (proper or improper), mixed numbers, and decimals. Fractions may be positive or negative. Decimals may be positive or negative and are limited to the thousandths place.

1.04 Real-world problems with rational numbers

Ideas

Real-world problems with rational numbers