32.1: Write Expressions Where Letters Stand for Numbers

Lesson

Let's use expressions with variables to describe situations.

Exercise \(\PageIndex{1}\): Algebra Talk: When \(x\) is \(6\)

If \(x\) is \(6\), what is:

\(x+4\)

\(7-x\)

\(x^{2}\)

\(\frac{1}{3}x\)

Exercise \(\PageIndex{2}\): Lemonade Sales and Heights

1. Lin set up a lemonade stand. She sells the lemonade for $0.50 per cup.
   1. Complete the table to show how much money she would collect if she sold each number of cups.

      Table \(\PageIndex{1}\)

      lemonade sold (number of cups)	\(12\)	\(183\)	\(c\)
      money collected (dollars)

   2. How many cups did she sell if she collected $127.50? Be prepared to explain your reasoning.
2. Elena is 59 inches tall. Some other people are taller than Elena.
   1. Complete the table to show the height of each person.

      Table \(\PageIndex{2}\)

      person	Andre	Lin	Noah
      how much taller than Elena (inches)	\(4\)	\(6\frac{1}{2}\)	\(d\)
      person's height (inches)

   2. If Noah is \(64\frac{3}{4}\) inches tall, how much taller is he than Elena?

Exercise \(\PageIndex{3}\): Building Expressions

1. Clare is 5 years older than her cousin.
   1. How old would Clare be if her cousin is:
      10 years old?
      2 years old?
      \(x\) years old?
   2. Clare is 12 years old. How old is Clare’s cousin?
2. Diego has 3 times as many comic books as Han.
   1. How many comic books does Diego have if Han has:
      6 comic books?
      \(n\) books?
   2. Diego has 27 comic books. How many comic books does Han have?
3. Two fifths of the vegetables in Priya’s garden are tomatoes.
   1. How many tomatoes are there if Priya’s garden has:
      20 vegetables?
      \(x\) vegetables?
   2. Priya’s garden has 6 tomatoes. How many total vegetables are there?
4. A school paid $31.25 for each calculator.
   1. If the school bought \(x\) calculators, how much did they pay?
   2. The school spent $500 on calculators. How many did the school buy?

Are you ready for more?

Kiran, Mai, Jada, and Tyler went to their school carnival. They all won chips that they could exchange for prizes. Kiran won \(\frac{2}{3}\) as many chips as Jada. Mai won 4 times as many chips as Kiran. Tyler won half as many chips as Mai.

1. Write an expression for the number of chips Tyler won. You should only use one variable: \(J\), which stands for the number of chips Jada won.
2. If Jada won 42 chips, how many chips did Tyler, Kiran, and Mai each win?

Summary

Suppose you share a birthday with a neighbor, but she is 3 years older than you. When you were 1, she was 4. When you were 9, she was 12. When you are 42, she will be 45.

If we let \(a\) represent your age at any time, your neighbor’s age can be expressed \(a+3\).

your age	\(1\)	\(9\)	\(42\)	\(a\)
neighbor's age	\(4\)	\(12\)	\(45\)	\(a+3\)

Table \(\PageIndex{3}\)

We often use a letter such as \(x\) or \(a\) as a placeholder for a number in expressions. These are called variables (just like the letters we used in equations, previously). Variables make it possible to write expressions that represent a calculation even when we don't know all the numbers in the calculation.

How old will you be when your neighbor is \(32\)? Since your neighbor's age is calculated with the expression \(a+3\), we can write the equation \(a+3=32\). When your neighbor is \(32\) you will be \(29\), because \(a+3=32\) is true when \(a\) is \(29\).

Practice

Exercise \(\PageIndex{4}\)

Instructions for a craft project say that the length of a piece of red ribbon should be 7 inches less than the length of a piece of blue ribbon.

1. How long is the red ribbon if the length of the blue ribbon is:
   10 inches?
   27 inches?
   \(x\) inches?
2. How long is the blue ribbon if the red ribbon is 12 inches?

Exercise \(\PageIndex{5}\)

Tyler has 3 times as many books as Mai.

1. How many books does Mai have if Tyler has:
   15 books?
   21 books?
   \(x\) books?
2. Tyler has 18 books. How many books does Mai have?

Exercise \(\PageIndex{6}\)

A bottle holds 24 ounces of water. It has \(x\) ounces of water in it.

1. What does \(24-x\) represent in this situation?
2. Write a question about this situation that has \(24-x\) for the answer.

Exercise \(\PageIndex{7}\)

Write an equation represented by this tape diagram using each of these operations.

Figure \(\PageIndex{1}\)

1. addition
2. subtraction
3. multiplication
4. division

(From Unit 6.1.1)

Exercise \(\PageIndex{8}\)

Select all the equations that describe each situation and then find the solution.

1. Han's house is 450 meters from school. Lin’s house is 135 meters closer to school. How far is Lin’s house from school?
   • \(z=450+135\)
   • \(z=450-135\)
   • \(z-135=450\)
   • \(z+135=450\)
2. Tyler's playlist has 36 songs. Noah’s playlist has one quarter as many songs as Tyler's playlist. How many songs are on Noah’s playlist?
   • \(w=4\cdot 36\)
   • \(w=36\div 4\)
   • \(4w=36\)
   • \(\frac{w}{4}=36\)

(From Unit 6.1.4)

Exercise \(\PageIndex{9}\)

You had $50. You spent 10% of the money on clothes, 20% on games, and the rest on books. How much money was spent on books?

(From Unit 3.4.3)

Exercise \(\PageIndex{10}\)

A trash bin has a capacity of 50 gallons. What percentage of its capacity is each amount? Show your reasoning.

1. 5 gallons
2. 30 gallons
3. 45 gallons
4. 100 gallons

(From Unit 3.4.5)