A recreation area is proposing an increase in the rates for kayak rentals. The initial charge is $32.00 and increase will be $35.00. Kayak rental is $5.00(kayak) and the increase will be $7.00. ...

A recreation area is proposing an increase in the rates for kayak rentals. The initial charge is $32.00 and increase will be $35.00. Kayak rental is $5.00(kayak) and the increase will be $7.00.  Write a variable expression showing the total cost to rent a kayak at the current  rate. Write a variable expression showing the total cost to rent a kayak at the increased rate.

Use the variable expressions to calculate the cost to rent a kayak for 8 hours at the current rate and the increased rate.

Calculate the percent increase in the total cost to rent a kayak for 8 hours if the proposed rate if the proposed rate is approved.

Emma paid $62.00 for a kayak rental at the current rate. Write and solve an equation to determine the number of hours she rented the kayak.

Asked on by taylorruth

2 Answers | Add Yours

caledon's profile pic

caledon | High School Teacher | (Level 3) Senior Educator

Posted on

An important piece of information that was not included is the per-hour rate. I'm assuming that this is what the $5 and $7 charges apply to.

Writing variable equations:

The initial charge represents a starting amount greater than zero; since there is no way of reducing charges, the equation must begin at the value of this initial charge. The per-hour rate is also fixed, so the real variable in this equation in the number of hours for which the kayak is rented. This number (the hours) is multiplied by the per-hour rate, and the product is added to the initial charge.

Initial charge + (per-hour charge)hours rented

Current rate: 32 + 5x

Increased rate: 35 + 7x

Renting a kayak for 8 hours:

To solve this, we only need to plug in the value of 8 for x:

5(8) = 40     Current rate: 32 + 40 = 72

7(8) = 56     Increased rate: 35 + 56 = 91

Percent increase in an 8 hour period:

Another way of asking this is "how many times does 72 go into 91?" The answer will clearly be more than one. Multiplying this by 100 will tell us the percentage relationship between the two.

(91/72) x 100 = 126.39%

This does not mean the increase is 126.39%. If 10 was increased by 100%, that would equal 20. The increase is actually 26.39%; 100% of the original 72, plus an additional 26.39% of 72.

100% of 72 = 72

26.39% of 72 = 19

72 + 19 = 91

$62.00 rental:

Our equation for rentals at the current rate is 32 + 5x = total.

Emma's total is 62.

First, eliminate 32 from both sides:  5x = 30

Then divide by 5: x = 6

Emma rented the kayak for 6 hours.

The full equation would read (62-32)/5 = x

Alternately, we can just find the difference between 62 and 72, the time for an 8-hour rental. This is a value of 10, and since we know it's $5 per hour, she must have rented it for 2 hours less than 8 hours.

Sources:
maria-vivanco's profile pic

maria-vivanco | Student, Grade 11 | (Level 1) Valedictorian

Posted on

As the person above mentioned, the per-hour rate isn't mentioned however we can assume that is what the $5 and $7 are for.

The expressions are expressed such as:

the initial amount + the per-hour rate(amount of hours rented)

For the current rate it would be:

32 + 5x

(x being the amount of hours the kayak was rented) 

For the increased rate:

35 + 7x

To find how much it would cost if you rented it for 8 hours you would substitute 8 for x

for the current rate:  32+5(8)= 32+40= 72

For the increased rate: 35 + 7(8)= 35+56= 91

Percent Increase: 

1. you can take the increased cost for 8  hours (which is 91) and find the difference between the two costs (the current and increased) 

91-72= 19

That is how much it increased. to find the percent you want to know what percent is 19 of 72. to find that out, you would do : 19/72= .2638*100= 26.38% 

Emma:

To find how many hours Emma rented the kayak if she paid 62, you would make the expression (the current one) to equal 62

32 + 5x= 62

subtract both sides by 32

5x=62-32

5x=30

divide both sides by 5

x=6

Emma rented the kayak for 6 hours. 

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