A retailer purchases an item at a wholesale price. The retailer marks up the wholesale pri… — SAT

A retailer purchases an item at a wholesale price. The retailer marks up the wholesale price by $p\%$ to establish the retail price. During a clearance sale, the retailer discounts the retail price by $(p - 10)\%$ . If the clearance sale price of the item is $8\%$ greater than the original wholesale price, and $p > 10$ , what is the value of $p$ ?

Cevap: 20

Cevap

The value of

p

20

The correct answer is

20

. By representing the markup and discount as decimal multipliers, we can write the equation for the final price as a function of the wholesale price:

W \left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right) = 1.08W

. Dividing by

W

and letting

y = \frac{p}{100}

, we get

(1 + y)(1.1 - y) = 1.08

. Expanding this gives

1.1 + 0.1y - y^2 = 1.08

, which rearranges to the quadratic equation

y^2 - 0.1y - 0.02 = 0

. Factoring this equation yields

(y - 0.2)(y + 0.1) = 0

. Since

p > 10

y

must be positive, which means

y = 0.2

. Therefore,

p = 20

Adım Adım Çözüm

Express the retail price in terms of the wholesale price

W

and the markup percentage

p\%

Retail Price =

W \left(1 + \frac{p}{100}\right)

A markup of

p\%

increases the base price

W

by a factor of

\left(1 + \frac{p}{100}\right)

Express the clearance sale price after applying a discount of

(p - 10)\%

to the retail price.

Clearance Price =

W \left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right)

A discount of

(p - 10)\%

decreases the retail price by a factor of

\left(1 - \frac{p - 10}{100}\right)

Set the clearance price equal to

1.08W

, which represents an

8\%

increase over the wholesale price, and simplify the equation by dividing both sides by

W

\left(1 + \frac{p}{100}\right)\left(1 - \frac{p - 10}{100}\right) = 1.08

The final clearance price is

8\%

greater than the wholesale price

W

, so we equate it to

1.08W

and divide both sides by

W

to eliminate the variable.

Substitute

y = \frac{p}{100}

into the simplified equation and expand the terms.

(1 + y)(1.1 - y) = 1.08 \Rightarrow 1.1 + 0.1y - y^2 = 1.08

Writing the equation in terms of

y

simplifies the algebraic expansion. The term

1 - \frac{p - 10}{100}

becomes

1 - (y - 0.1) = 1.1 - y

Rearrange the quadratic equation into standard form, factor it, and solve for

y

y^2 - 0.1y - 0.02 = 0 \Rightarrow (y - 0.2)(y + 0.1) = 0 \Rightarrow y = 0.2

(since

p > 10

y > 0.1

)

Factoring the quadratic yields

y = 0.2

and

y = -0.1

. Since

p > 10

y

must be positive, which leaves

y = 0.2

as the only valid solution.

Convert the value of

y

back to

p

p = 20

Since

y = \frac{p}{100} = 0.2

, multiplying both sides by

100

gives

p = 20

Anahtar Kavram

Compounding percent changes algebraically using variable markups and discounts.

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