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How to Formulate an Equation Representing a Problem

Below are examples where you have to formulate an equation yourself, out of a word problem, in order to solve the exercise.

Sorting and interpreting the information from the text of a word problem can be difficult. Reading about problem solving techniques will also be helpful.

Example 1

Albert Einstein is 101 years older than Sheldon Cooper. Stephen Hawking is 38 years older than Sheldon Cooper. Added together, the three are 250 years old. How old are Albert Einstein, Sheldon Cooper and Stephen Hawking?

First, write down the sentences you can turn into equations. Then, translate them into equations:

Einstein is $101$ years older than Cooper.

Einstein is $101$ plus Cooper:

E = 101 + C

Hawking is $38$ years older than Cooper.

Hawking is $38$ plus Cooper:

H = 38 + C

Added together, they’re $250$ years old.

Einstein plus Cooper plus Hawking is $250$ years:

E + C + H = 250

To solve this, you can just insert the expressions for $E$ and $H$ into the third equation:

\begin{array}{l} E + C + H & = 250 \\ 101 + C + 38 + C + C & = 250 \\ 3 C & = 250 - 101 - 38 \\ \frac{3 C}{3} & = \frac{111}{3} \\ C & = 37 \end{array}

\begin{array}{l} E + C + H & = 250 \\ 101 + C + 38 + C + C & = 250 \\ 3 C & = 250 - 101 - 38 \\ \frac{3 C}{3} & = \frac{111}{3} \\ C & = 37 \end{array}

That means Sheldon Cooper is

37

years old, which makes Albert Einstein

37 + 101 = 138

years old and makes Stephen Hawking

37 + 38 = 75

years old.

Example 2

Four adult tickets and two children’s tickets to a Beyoncé concert cost $$ 440$ . A group of friends buy four children’s tickets and one adult ticket to the same concert, and they pay $$ 320$ in total. What is the price of one adult ticket, and what is the price of one children’s ticket?

The first thing you have to do with word problems is to write each sentence as an equation. Look at this:

Item 1:

Four adult tickets and two children’s tickets cost $ $440$ . If you know the price of an adult ticket $x$ and the price of a children’s ticket for $y$ , you can translate this to:

4 x + 2 y = 440 .

Item 2:

One adult ticket and four children’s tickets cost $ $320$ . You can translate this into:

x + 4 y = 320 .

Then you have two equations with two variables. That means you can solve them by either using the substitution method, or the elimination method. Here, I’ve used the substitution method, because that’s my favorite.

\begin{array}{l} (1) 4 x + 2 y & = 4400 \\ (2) x + 4 y & = 3200 \\ x & = 3200 - 4 y \end{array}

\begin{array}{l} (1) 4 x + 2 y & = 4400 & (2) x + 4 y & = 3200 \\ x & = 3200 - 4 y \end{array}

Substituting (2) into (1) gives:

\begin{array}{l} 4 (320 - 4 y) + 2 y & = 440 \\ 1280 - 16 y + 2 y & = 440 \\ - 14 y & = 440 - 1280 \\ \frac{- 14 y}{- 14} & = \frac{- 840}{- 14} \\ y & = 60 \end{array}

Inserting this value back into (2):

\begin{array}{l} x & = 320 - 4 \cdot 60 \\ x & = 320 - 240 \\ x & = 80 \end{array}

The price of an adult ticket is $ $80$ , and the price of a children’s ticket is $ $60$ .

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