- c.Let x= the length of the first piece. Then, 2x– 1 = the length of the second piece and 3(2x– 1) + 10 =
the length of the third piece. The sum of the lengths of these three smaller pieces will be the length of
the original piece of rope. This is represented as the equation x+ (2x– 1) + 3 (2x– 1) + 10 = 60. To
solve this equation, we first simplify the left side to obtain 9x+ 6 = 60. Solving this equation gives us x =
- Therefore, the length of the first piece is 6 feet, the second piece is 11 feet, and finally, the third piece is
43 feet long. So, we conclude that the longest piece of rope is 43 feet long.
- b.Let x = number of canisters of Ace balls. Then,x+ 1 = number of canisters of Longline balls. The
important observation is that multiplying the price of one canister of Ace balls by the number of canis-
ters of Ace balls results in the portion of the total amount spent on Ace balls. The same reasoning is true
for the Longline balls. So, we must solve an equation of the form:
amount spent on Ace balls + amount spent on Longline balls = total amount spent
Using the information provided, this equation becomes 3.50x+ 2.75(x+ 1) = 40.25. Simplifying the left
side of the equation yields 6.25x+ 2.75 = 40.25. Subtracting 2.75 from both sides and then dividing by
6.25 yields the solution x = 6. So, we conclude that he bought 6 canisters of Ace balls and 7 canisters of
Longline balls.
- d.Let x = the number of gallons needed of the 30% nitrogen. Then, since we are supposed to end up
with 10 gallons, it must be the case that 10 – x= the number of gallons needed of the 90% nitrogen.
Multiplying the number of gallons of 30% nitrogen by its concentration yields the amount of nitrogen
contained within the 30% solution. A similar situation holds for the 90% nitrogen, as well as for the
final 70% solution. We must solve an equation of the following form:
amount of nitrogen contributed + amount of nitrogen contributed = total amount of nitrogen
from the 30% solution from the 90% solution in the entire 10 gallons
Using the information provided, this equation becomes 0.30x+ 0.90(10 – x) = 0.70(10), which is solved
as follows:
0.30x+ 0.90(10x– x) = 0.70(10)
30 x+ 90(10 – x) = 70(10)
30 x+ 900 – 90x) = 700
–60x+ 900 = 700
–60x= –200
x= ––^260 00 = ^130
Thus, rounding to two decimal places, we conclude that she should mix approximately 3.33 gallons of
the 30% nitrogen solution with 6.67 gallons of the 90% nitrogen solution to obtain the desired mixture.
ANSWERS & EXPLANATIONS–
