algebra 1b unit 4

Question

Henry walked to school at 2.5 mph the first morning. He wanted to meet his friends and have 6 minutes before the bell to play. He arrived just as the bell rang, so the next day, he decided to increase his speed to 3.5 mph and he got there with 7 minutes to play. How many miles is Henry’s school from his house? Do not round your answer.

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Answer 1

Hey Serena!

The point of these types of word problems is usually to get you thinking about how to apply what you know about solving liner equations or simultaneous equations.

First let's think about how we can represent the distance, lets call it "d" from home to school.

The formula for distance is given as speed mph * time in hours = distance.

now we don't know distance or time but we can write the following from the first bit of information we are given using "t" for time and "d" for distance

2.5 mph*t = d

Now since we have two unknowns we need a second equations to fully solve. Aha let's use the second trip Henry took to write our second equation.

3.5 mph*(t-7) = d

notice our new speed is 3.5 mph and distance stays the same (same distance to school)

look at time t however... it took henry 7 less minutes to get to school compared to his last trip....but we need to still use the same variable t....thus we will write the second time as (t - 7)!

now we have our two equations and two unknowns:

1) 2.5 mph* t = d

2) 3.5mph*(t - 7 ) = d

Notice for both equations d is by itself, thus we can use substitution very easily to get an equation with only t.

substituting equation 1) into 2) we write:

3) 2.5 mph * t = 3.5 mph *(t-7)

(Quick note about units... speed is in miles per hour, time is in minutes, we will come back to this in a second)

now we solve equation 3, first multiply out the 3.5 (Also notice mph can be canceled out by division)

2.5t = 3.5t - 24.5

Add 24.5 to both sides:

24.5 + 2.5t = 3.5t, move 2.5t over by subtraction:

24.5 = t

now we have our time in MINUTES

it took Henry 24.5 minutes at 2.5 mile per hour.

The question is asking for the distance to school however!

recall our initial equation:

2.5mph*t = distance

We have t now so we can multiply to solve for distance.

HOWEVER we cannot multiply miles per hour x minutes.

We need to convert either mph to mile per minute, OR convert 24.5 minutes to hours in order to solve distance in miles.

Lets convert with the second method:

24.5 minutes *1 hour/60 minutes

= 0.40333 hours.

now we can solve correctly:

distance = 2.5 mph * .40333 hours = 1.020833 miles

if you like you can check this answer using the second distance equations:

3.5mph *(t-7) = d

Hope that helped!