Lesson 3

Using Equations to Solve Problems

Let’s use equations to solve problems involving proportional relationships.

3.1: Number Talk: Quotients with Decimal Points

Without calculating, order the quotients of these expressions from least to greatest.

\(42.6 \div 0.07\)

\(42.6 \div 70\)

\(42.6 \div 0.7\)

\(426 \div 70\)

Place the decimal point in the appropriate location in the quotient: \(42.6 \div 7 = 608571\)

Use this answer to find the quotient of one of the previous expressions.

3.2: Concert Ticket Sales

A performer expects to sell 5,000 tickets for an upcoming concert. They want to make a total of $311,000 in sales from these tickets.

  1. Assuming that all tickets have the same price, what is the price for one ticket?
  2. How much will they make if they sell 7,000 tickets?
  3. How much will they make if they sell 10,000 tickets? 50,000? 120,000? a million? \(x\) tickets?
  4. If they make $404,300, how many tickets have they sold?
  5. How many tickets will they have to sell to make $5,000,000?

3.3: Recycling

Aluminum cans can be recycled instead of being thrown in the garbage. The weight of 10 aluminum cans is 0.16 kilograms. The aluminum in 10 cans that are recycled has a value of $0.14.

  1. If a family threw away 2.4 kg of aluminum in a month, how many cans did they throw away? Explain or show your reasoning.
  2. What would be the recycled value of those same cans? Explain or show your reasoning.
  3. Write an equation to represent the number of cans \(c\) given their weight \(w\).
  4. Write an equation to represent the recycled value \(r\) of \(c\) cans.
  5. Write an equation to represent the recycled value \(r\) of \(w\) kilograms of aluminum.


The EPA estimated that in 2013, the average amount of garbage produced in the United States was 4.4 pounds per person per day. At that rate, how long would it take your family to produce a ton of garbage? (A ton is 2,000 pounds.)

Summary

Remember that if there is a proportional relationship between two quantities, their relationship can be represented by an equation of the form \(y = k x\). Sometimes writing an equation is the easiest way to solve a problem.

For example, we know that Denali, the highest mountain peak in North America, is 20,310 feet above sea level. How many miles is that? There are 5,280 feet in 1 mile. This relationship can be represented by the equation

\(\displaystyle f=5,\!280 m\)

where \(f\) represents a distance measured in feet and \(m\) represents the same distance measured in miles. Since we know Denali is 20,310 feet above sea level, we can write

\(\displaystyle 20,\!310=5,\!280 m\)

So \(m = \frac{20,310}{5,280}\), which is approximately 3.85 miles.

Glossary Entries

  • constant of proportionality

    In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

    In this example, the constant of proportionality is 3, because \(2 \boldcdot 3 = 6\), \(3 \boldcdot 3 = 9\), and \(5 \boldcdot 3 = 15\). This means that there are 3 apples for every 1 orange in the fruit salad.

    number of oranges number of apples
    2 6
    3 9
    5 15
  • proportional relationship

    In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.

    For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.

    We can write this relationship as \(p = 4s\). This equation shows that \(s\) is proportional to \(p\).

    \(s\) \(p\)
    2 8
    3 12
    5 20
    10 40