Question

Andrea is at the grocery store buying fruit for a smoothie stand. She needs 7 boxes of raspberries for every 5 bunches of bananas she buys. The price of a box of raspberries is $2.50 and the price of a bunch of bananas is $3.00. The sales tax is 8%. Andrea has $135.00 to spend on fruit. What is the greatest number of boxes of raspberries and the greatest number of bunches of bananas Andrea can buy if the ratio of boxes of raspberries to bunches of bananas is 7:5? Be sure to show your work.

Answer 1

Answer:

The greatest number of bunches of bananas that Andrea can buy is 15 and the greatest number of boxes of raspberries that Andrea can buy is 21 to maintain the ratio 7:5

Step-by-step explanation:

Let

x ---->the greatest number of boxes of raspberries that Andrea can buy

y ---->the greatest number of bunches of bananas that Andrea can buy

we know that

$100\%+8\%=106\%=108/100=1.08$ ---> sales tax

$\frac{x}{y} =\frac{7}{5}$

$x=\frac{7}{5}y$ ----> equation A

$1.08[2.50x+3.00y]\leq 135$ ----> inequality B

substitute equation A in the inequality B

$1.08[2.50(\frac{7}{5}y)+3.00y]\leq 135$

solve for y

Multiply by 5 both sides to remove the fraction

$18.9y+16.2y\leq 675$

$35.1y\leq 675$

$y\leq 19.23$

so

The maximum value of y is 19

Find the value of x

replace the value of y in equation A until you get an integer value that satisfies the ratio 7:5  

For y=19

$x=\frac{7}{5}(19)$

$x=26.6$

For y=18

$x=\frac{7}{5}(18)$

$x=25.2$

For y=17

$x=\frac{7}{5}(17)$

$x=23.8$

For y=16

$x=\frac{7}{5}(16)$

$x=22.4$

For y=15

$x=\frac{7}{5}(15)$

$x=21$

therefore

The greatest number of of bunches of bananas that Andrea can buy is 15 and the greatest number of boxes of raspberries that Andrea can buy is 21 to maintain the ratio 7:5

Verify the inequality B

$1.08[2.50(21)+3.00(15)]\leq 135$

$1.08[52.5+45]\leq 135$

$105.3\leq 135$ ----> is true