# Solving Elapsed Time Problems

Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Performing calculations for an event or task that has occurred in the past or will occur in the future, or figuring out how much time has passed, can be a very useful skill. Fortunately, there are a few rules that can make such calculations foolproof, as you'll find out in this lesson.

## A Problem with Time

You'd like to go to a concert tonight, but you need your parents' approval first. They're willing to let you go, as long as you finish your 20 minutes of nightly reading, practice for your piano lesson and wash the dishes.

Right now, it's 6:45 PM, and your friend is coming to pick you up at 7:30 PM. In addition to the 20 minutes of reading, you're pretty sure you know how long it's going to take you to practice the piano and wash the dishes. But you're not sure if you'll be able to finish everything by 7:30 PM. Not to worry, what we're looking at here is a very straightforward elapsed time problem that involves units of time, like hours and minutes.

## Units of Time

We call the time it takes the Earth to orbit around the sun a year. We call the time it takes the Earth to rotate once around its axis a day. We schedule our days according to the familiar units of hours, minutes and seconds. Additionally, we group years into progressively larger groups, like a decade (ten years), century (100 years) and millennium (1,000 years).

Elapsed time problems can work with any or all of these units of time. Let's take a look at how to solve these types of problems.

## Elapsed Time Equation

The equation for solving elapsed time problems could not be easier: T = L - E. Here:

T = Time elapsed

L = Later time

E = Earlier time

To make our elapsed time problems even easier to solve, we can also use our basic algebra skills to rearrange this equation, keeping in mind that the letters represent the same variable, no matter where they are in the equation. So, L = T + E and E = L - T.

We now have three equations that can help us solve any elapsed time problems. Let's take a look at a couple of examples.

## Problem-Solving Process

To solve the elapsed time problems below, we'll use the following three-step process:

1. Identify which variables are known.
2. Identify which equation to use.
3. Plug in the numbers and solve.

## Solving for T

The Buffalo Bills have not made the playoffs since 1999. It is now 2015. How long has it been since the Buffalo Bills made the playoffs?

1. The later time is 2015 (L), and the earlier time is 1999 (E).

2. Since we know L and E, we need to find T. This makes sense because we're trying to find the time elapsed since the last Bills' playoff appearance. Therefore, we'll use the equation: T = L - E.

3. T = 2015 - 1999.

4. T = 16 years. It has been 16 years since the Bills have been in a playoff game (sigh).

## Solving for L

Let's revisit that concert you'd like to attend. You've estimated that reading will take 20 minutes, piano will take 10 minutes and the dishes will take 30 minutes. Will you be ready by 7:30 PM?

1. We know that time elapsed is just the three activities added together: 20 + 10+ 30 = 60 minutes (T). We also know that the earlier time is 6:45 PM (E).

2. Since we know T and E, and we are trying to find L, we'll use the equation: L = T + E.

3. L = 6:45 + 60 minutes.

4. L = 7:45 PM. Rats! You won't be ready for the concert by 7:30 PM. You'll be done with your chores by 7:45 PM. Perhaps you can ask your friend to wait.

## Solving for E

Your friend watched a movie that ended at 10:10 PM. The movie is two hours and five minutes long. What time did she start watching the movie?

1. The later time is 10:10 PM (L), and the elapsed time is 2 hours and 5 minutes, or 2:05 (T).

2. Since we know L and T, we're trying to find E. We'll use the equation: E = L - T.

3. E = 10:10 PM - 2:05.

4. I = 8:05 PM. The movie started at 8:05 PM.

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