a landscaper bought some decorative cement blocks from a landsccaping supplier. the supplier charged 5% sales tax and the total came to $315. without the tax, the landscaper could have bought 6 more blocks for the same total cost. how many blocks did the landscaper buy?

Let's break down the problem step by step:

Step 1: Let's assume the cost of each decorative cement block is x dollars.
Step 2: The total cost without sales tax can be calculated as the cost per block multiplied by the number of blocks: x * n, where n is the number of blocks.
Step 3: The sales tax is 5% of the total cost without tax, which means the sales tax amount is 5/100 * (x * n).
Step 4: The total cost including sales tax is the total cost without tax plus the sales tax amount. This can be written as:
x * n + (5/100) * (x * n)
= x * n + (5/100) * x * n
= x * n * (1 + 5/100)

Step 5: From the problem statement, we know that the total cost including sales tax is $315. Thus, we can set up the equation:
x * n * (1 + 5/100) = 315

Step 6: We also know that without the sales tax, the landscaper could have bought 6 more blocks for the same total cost. So, the total cost without sales tax is $(315 - 6x) and the number of blocks is (n + 6).

Step 7: Using the total cost without sales tax and the number of blocks, we can write the equation:
x * (n + 6) = 315 - 6x

Step 8: Now we have a system of two equations with two variables:
x * n * (1 + 5/100) = 315 (Equation 1)
x * (n + 6) = 315 - 6x (Equation 2)

Step 9: We can solve this system of equations to find the values of x and n. Let's solve it using the substitution method:

From Equation 2:
x * (n + 6) = 315 - 6x
xn + 6x = 315 - 6x
xn = 315 - 12x
n = (315 - 12x) / x

Substituting this value of n into Equation 1:
x * (315 - 12x) / x * (1 + 5/100) = 315

Step 10: Simplifying the equation:
(315 - 12x) / (1 + 5/100) = 315
(315 - 12x) / (1 + 0.05) = 315
(315 - 12x) / 1.05 = 315

Step 11: Multiplying both sides of the equation by 1.05 to eliminate the denominator:
315 - 12x = 315 * 1.05
315 - 12x = 330.75

Step 12: Solving for x:
-12x = 330.75 - 315
-12x = 15.75
x = 15.75 / -12
x = -1.3125

Since the cost of each block cannot be negative, we can conclude that there is an error in the problem statement or the given information.

Please check the problem statement or provide additional information if possible.

To solve this problem, we can use algebraic equations to represent the given information.

Let's start by assuming the cost of each decorative cement block is "x" dollars.

Without the sales tax, the total cost of the blocks would be the cost per block multiplied by the number of blocks:

Total cost without tax = x * number of blocks

Now, considering the sales tax of 5%, we need to add the tax to the total cost. The tax amount is calculated by multiplying the total cost without tax by 5% (or 0.05):

Tax amount = 0.05 * (x * number of blocks)

So, the total cost including the sales tax is:

Total cost = Total cost without tax + Tax amount

Given that the total cost is $315, we can set up the following equation:

315 = x * number of blocks + 0.05 * (x * number of blocks)

Now, let's consider the second part of the problem. If the landscaper had not paid tax, they could have bought 6 more blocks for the same total cost. This means the total cost without tax would remain the same. Therefore, we write:

315 = x * (number of blocks + 6)

Now we have a system of two equations:

315 = x * number of blocks + 0.05 * (x * number of blocks)
315 = x * (number of blocks + 6)

To solve this system of equations, we need to find the value of "x" and the number of blocks ("number of blocks").

Let's solve this using the elimination or substitution method.

From the first equation, we can simplify it:

315 = x * (1 + 0.05) * number of blocks
315 = 1.05 * x * number of blocks

Now, we have the following equation:

315 = 1.05 * x * number of blocks

Next, we solve the second equation:

315 = x * (number of blocks + 6)

We can rearrange this equation:

number of blocks + 6 = 315 / x

Subtract 6 from both sides:

number of blocks = (315 / x) - 6

Now, substitute this expression for "number of blocks" in the simplified equation:

315 = 1.05 * x * ((315 / x) - 6)

Simplify further:

315 = 1.05 * (315 - 6x)

Expand:

315 = 330.75 - 6.3x

Rearrange this equation:

6.3x = 330.75 - 315

6.3x = 15.75

Divide both sides by 6.3 to isolate x:

x = 15.75 / 6.3

x = 2.5

Now substitute the value of x back into the expression for "number of blocks":

number of blocks = (315 / 2.5) - 6

number of blocks = 126 - 6

number of blocks = 120

Therefore, the landscaper bought 120 blocks.

If there were n bricks at price p, then we know that

1.05np = 315
(n+6)p = 315

Now just eliminate p and solve for n.

number of blocks --- x

cost per block --- y
1.05xy = 315
xy = 300

without sales tax the cost would have been

(x+6)y = 315

so divide the two equations:
( (x+6)y)/(xy) = 315/300
(x+6)/x = 21/20
21x = 20x + 120
x = 120 , then y = 2.5

He bought 120 blocks at $2.50 each

check:
120(2.5)(1.05) = 315
with no tax:
126(2.5) = 315 , he could have bought 6 more, yeahh