The cost of a taxi ride is \$2 for the first 1/5 mile and \$.30 for each additional 1/5 mile. From 8pm to 6am there is a \$.50 surcharge added to the cost of the ride after the first 1/5 mile. If Elizabeth got into a taxi at 9:05pm and paid \$7.60, how far did she ride to the nearest 1/5 mile?

Relevance

Sounds like an algebra problem but there's no need for me to be picky. ;)

Since Elizabeth spent a lot of money, let's assume she traveled more than just 1/5 mile.

First lets consider the initial cost of the ride. \$2 for the first 1/5 mile. But since Elizabeth gets into the taxi at around 9pm, this is between the 8pm to 6am period where there is an extra \$0.50 surcharge. Therefore the cost for the first 1/5 mile is \$2.50

Next we'll use the letter N to represent the number of "1/5 miles" she traveled after the first 1/5 mile.

Every 1/5 mile after the first costs Elizabeth \$0.30. To determine how much she spends after the first 1/5 mile, all we have to do is multiply the number of 1/5 miles after the first (which we decided to call "N") by the price. So that part of the trip costs Elizabeth \$0.30 x N.

If we add what she paid for the first 1/5 mile to what she paid for the rest of the ride, we get the total cost of the ride, \$7.60.

Mathematically speaking:

\$First 1/5 + \$Rest of trip = \$Total

\$2.50 + \$0.30 x N = \$7.60

Now solve the equation for N. Subtract \$2.50 from both sides.

\$0.30 x N = \$5.10

Divide both sides by \$0.30

N=17

That is 17 "1/5 miles" that she travels AFTER the first 1/5 mile. Therefore the total distance she travels is 18 "1/5 miles" if we add the first 1/5 mile in.

I assume your teacher doesn't want 18 "1/5 miles" as an answer so we'll need a numerical value. Just multiply 18 by 1/5 to get a total number of "whole" miles.

18 x 1/5

=3.6 miles

=3 and 3/5 miles

Well I'm guessing that in the time it took me to type all that, two or three other people have probably already answered. Oh well.

Edit: It also seems that there is confusion about whether to add the surcharge for every 1/5 mile after the first or just to the total cost. Your problem statement says "there is a \$.50 surcharge added to the <i>cost of the ride</i>" so I assume it's just the total cost. If I am right, your answer is 3 and 3/5. If I am wrong, your answer is 1 and 2/5 like so many other people have suggested.

• First thing you need to do is break apart the equation and define variables. Look for what's important in the word problem.

The distance is unknown so use a variable

The total 1/5's after the 1st mile= x

The rates

rate 1 = \$2 for the 1st 1/5 of a mile

rate 2 = \$0.30 for every 1/5 mile AFTER the 1st one.

rate 3 = \$0.50 surcharge for every 1/5 mile EXCEPT the 1st one.

The total = \$7.60

First... do you use the surcharge?

Well Elizabeth got into the cab after 8pm so the answer is yes.

That means that you now have enough information to set up the equation.

2{The 1st 1/5} + x{remaining 1/5's} * (0.30 {Rate 2} + 0.50 {Rate 3}) = 7.60

Which cleaned up looks like this...

2 + x*(0.30 + 0.50) = 7.60

Combine what's in the parentheses...

2 + x*(0.80) = 7.60

Subtract 2 from each side...

2 - 2 + x*(0.80) = 7.60 - 2

then divide each side by x's coefficient...

x*(0.80) / 0.80 = 5.60 / 0.80

Gives you x...

x = 7

Check it to make sure it works by substituting seven into the problem for the variable x...

2 + 7*(0.80) = 7.60

2 + 5.60 = 7.60

7.60 = 7.60

It works.

That means that Elizabeth drove seven 1/5 miles or 1 & 2/5 miles.

• This is a real life example. Why not do it by mental math?

First, subtract the first 1/5 mile charge and the surcharge from the total \$7.60.

\$7.60 - \$2 - \$0.50 = \$5.1

Since each additional (1/5) mile charge is \$0.30, \$0.3 goes into \$5.1 17 times, which means the total miles should be 18(1/5) miles.

• the amount Elizabeth paid=\$7.60

less the night surcharge=\$7.10

less \$2 for the first 1/5 thmil2=\$5.10

divided by the tariff per mile=17

so her ride was for 18*1/5=3 3/5 mile

• ok first you subtracted the \$2 leaving \$5.60. Since it is after 8pm it is \$.80 a 1/5 mile. so divide the \$5.60 by \$.80 and you get...7. so She went 7/5 or 1 2/5 of a mile.

• 7.60 - 2.00 = 5.60 1/5 mile

5.60 - .50 = 5.10 1/5 + surcharge

5.10 - .30x = 0.00

.30x = -5.10

x = -17 (cant have negative mileage)

x = 17 1/5s of a mile

17 * .2 (1/5) = 3.4 3.4 extra miles

3.4 + .2 = 3.6 = 3 3/5 miles

she rode 3 3/5 miles