Gavin likes biking. This week his goal is to bike about 65 total miles over four days. Each day he wants to ride 1.5 times as far as he rode the day before. How far should he ride on each of the four days. Write the expressions for the distance Gavin needs to bike each day. Then relate these expressions to create an equation that represents Gavin's situation.

Step-by-step explanation:

let the first-day ride be x

Then the second-day ride will be =1.5 x

The third-day ride will be=1.5(1.5x)      

The Fourth-day ride will be=1.5*1.5(1.5x)  

hence for the four days the distance covered is

65=x+1.5x+1.5*1.5x+1.5*1.5*1.5x

65=x+1.5x+2.25x+3.375x

65=8.125x

divide both sides by 8.125 we have

x=65/8.125

x=8mile

the first-day ride be x=8miles

the second-day ride will be =1.5 x =1.5*8= 12miles

The third-day ride will be=1.5(1.5x) = 18miles

The Fourth-day ride will be=1.5*1.5(1.5x) = 27miles

The variable x represents the distance Gavin bikes on the first day.

On the second day, he wants to bike 1.5 times the distance he rode the first day, x. The expression for the distance he should bike on the second day is 1.5x.

On the third day, he wants to bike 1.5 times the distance he rode the second day, 1.5x. The expression for the distance he should bike on the third day is (1.5)(1.5x) = 2.25x.

On the fourth day, he wants to bike 1.5 times the distance he rode the third day, 2.25x. The expression for the distance he should bike on the fourth day is (1.5)(2.25x) = 3.375x.

The expressions for the distance Gavin should ride each of the following three days are 1.5x, 2.25x, and 3.375x.

The total distance Gavin should bike over the four days is the sum of the expressions corresponding to each day. So, the expression representing the total distance he should bike in the four days is x + 1.5x + 2.25x + 3.375x.

Gavin’s goal is to bike 65 miles in four days. The expression representing the total distance he should bike in four days is x + 1.5x + 2.25x + 3.375x.

His goal and the distance he should bike over the four days must be equal. So the equation is 65 = x + 1.5x + 2.25x + 3.375x. This equation can be used to find the number of miles Gavin should bike on the first day, x.

In the equation from part C, 65 = x + 1.5x + 2.25x + 3.375x, the four terms on the right-hand side—x, 1.5x, 2.25x, and 3.375x—are the like terms involving x.

The equation from question 1, part C, is 65 = x + 1.5x + 2.25x + 3.375x. The like terms involving x in this equation are x, 1.5x, 2.25x, and 3.375x.

Combine the like terms to get this equation.

65 = (1 + 1.5 + 2.25 + 3.375)x

65 = 8.125x

The simplified equation is 8.125x = 65.

The equation from part A is 8.125⁢x = 65.

Dividing both sides of the equation by the coefficient of x, 8.125, gives the following equation.

8.125⁢x/8.125 = 65/8.125

x =  65/8.125

x =  8

The value of x is 8.

The number of miles Gavin should ride the first day is x = 8.

The number of miles he should ride the second day is 1.5x = 1.5(8) = 12.

The number of miles he should ride the third day is 2.25x = 2.25(8) = 18.

And the number of miles he should ride the fourth day is 3.375x = 3.375(8) = 27.

So, the number of miles Gavin should ride on each of the four days is 8, 12, 18, and 27.

The result in part D confirms that the distances Gavin should cover on each day add up to 65 miles. So, the number of miles Gavin should ride each of the four days (8, 12, 18, and 27) is correct.