A linear equation models the total cost, in dollars, of purchasing x pounds of almonds and… — SAT

A linear equation models the total cost, in dollars, of purchasing $x$ pounds of almonds and $y$ pounds of walnuts. Under the original pricing, purchasing $8$ pounds of almonds and $15$ pounds of walnuts costs $c$ dollars. If the price per pound of almonds is increased by $25\%$ and the price per pound of walnuts is decreased by $10\%$ , the cost of purchasing $8$ pounds of almonds and $15$ pounds of walnuts is still $c$ dollars. Under the original pricing, a customer can purchase exactly $22$ pounds of almonds and no walnuts for $c$ dollars. How many pounds of walnuts and no almonds can the customer purchase for $c$ dollars under the original pricing?

Answer: 16.5 pounds

Answer

16.5

Under the original pricing, the cost of

8

pounds of almonds at

a

dollars per pound and

15

pounds of walnuts at

b

dollars per pound is represented by the linear equation

8a + 15b = c

. Under promotional pricing, the cost of almonds increases by

25\%

1.25a

, and the cost of walnuts decreases by

10\%

0.90b

. The new cost for the same amounts of nuts is

8(1.25a) + 15(0.90b) = 10a + 13.5b = c

. Setting the two cost equations equal to each other gives

8a + 15b = 10a + 13.5b

, which simplifies to

1.5b = 2a

, or

a = 0.75b

. Since

22

pounds of almonds can be purchased for

c

dollars, we have

22a = c

. Substituting

a = 0.75b

yields

22(0.75b) = c

, which simplifies to

16.5b = c

. Thus, exactly

16.5

pounds of walnuts can be purchased for

c

dollars.

Step-by-Step Solution

Set up the equation representing the total cost of the initial purchase under the original pricing.

8a + 15b = c

, where

a

is the original price per pound of almonds and

b

is the original price per pound of walnuts.

To represent the cost relation using the original individual prices of almonds and walnuts.

Modify the individual prices for the promotional rates and write the new total cost equation.

8(1.25a) + 15(0.90b) = c \implies 10a + 13.5b = c

The price per pound of almonds increases by

25\%

(multiplied by

1.25

) and the price per pound of walnuts decreases by

10\%

(multiplied by

0.90

Equate the two expressions representing

c

to find the ratio between the prices

a

and

b

8a + 15b = 10a + 13.5b \implies 2a = 1.5b \implies a = 0.75b

Since both purchasing combinations yield the same total budget

c

, their cost equations are equal.

Use the budget equation for purchasing only almonds to determine the equivalent purchase in walnuts.

22a = c \implies 22(0.75b) = c \implies 16.5b = c

. Therefore,

16.5

pounds of walnuts can be bought for

c

dollars.

Substituting the price relationship

a = 0.75b

allows expressing the budget

c

purely in terms of the price of walnuts

b

Key Concept

Modeling linear relationships in two variables and analyzing changes in coefficients.

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