Question

mandy’s candies is most famous for their homemade fudge and homemade toffee. they make an average of 154 pounds of fudge and toffee combined month. they sell a pound of fudge for $8.83 and a pound of toffee for $7.85. mandy’s averages $1,281.42 each month in fudge and toffee sales. on ave how many pounds each of fudge and toffee does mandy’s candies sell in one month?

a. 74 fudge; 80 toffee

b. 80 fudge; 74 toffee

c. 77 fudge; 77 toffee

d. 63 fudge; 91 toffee

Explanation:

Step1: Define Variables

Let \( f \) be pounds of fudge, \( t \) be pounds of toffee. We know \( f + t = 154 \) and \( 8.83f + 7.85t = 1281.42 \).

Step2: Test Option B

For B: \( f = 80 \), \( t = 74 \).
Check \( f + t = 80 + 74 = 154 \) (matches total).
Check sales: \( 8.83(80) + 7.85(74) = 706.4 + 580.9 = 1287.3 \)? Wait, no, wait original sales is 1281.42. Wait, maybe miscalculation. Wait, let's recalculate: \( 8.83\times80 = 706.4 \), \( 7.85\times74 = 7.85\times(70 + 4)=549.5 + 31.4 = 580.9 \). Sum: \( 706.4 + 580.9 = 1287.3 \). Wait, that's not 1281.42. Wait, maybe I made a mistake. Wait, let's check option A: \( f = 74 \), \( t = 80 \). \( 8.83\times74 + 7.85\times80 \). \( 8.83\times74 = 8.83\times(70 + 4)=618.1 + 35.32 = 653.42 \). \( 7.85\times80 = 628 \). Sum: \( 653.42 + 628 = 1281.42 \). Oh! I messed up t and f. So for A: \( f = 74 \), \( t = 80 \). Then \( 74 + 80 = 154 \), and sales: \( 8.83\times74 + 7.85\times80 = 653.42 + 628 = 1281.42 \), which matches. Wait, but earlier I thought B, but no. Wait, the problem says "how many pounds each of fudge and toffee". Let's re-express:

Let me set up equations properly. Let \( f \) = fudge, \( t \) = toffee.

Equation 1: \( f + t = 154 \) => \( t = 154 - f \)

Equation 2: \( 8.83f + 7.85t = 1281.42 \)

Substitute \( t = 154 - f \) into equation 2:

\( 8.83f + 7.85(154 - f) = 1281.42 \)

\( 8.83f + 1208.9 - 7.85f = 1281.42 \)

\( 0.98f = 1281.42 - 1208.9 = 72.52 \)

\( f = 72.52 / 0.98 = 74 \). Then \( t = 154 - 74 = 80 \). So fudge is 74, toffee is 80. So option A.

Wait, earlier when I tested B, I had f and t reversed. So correct is A: 74 fudge, 80 toffee.

Step3: Confirm with Equations

From equation solving, \( f = 74 \), \( t = 80 \) satisfies both \( f + t = 154 \) and \( 8.83f + 7.85t = 1281.42 \).

Answer:

A. 74 fudge; 80 toffee