1. Create "let statements" in order to solve the following algebraic word problem. Your solution MUST include the following: (a) Let statements (b) An algebraic equation and a solution (c) A check - substitute the value of the variable into the algebraic equation (d) Therefore statements. /5 marks]
Four students go shopping at Vaughan Mills. Each student brought a different amount of money to the mall. Chloe brought 5 more than half the amount that Luca brought. Nico brought 2 less than double the amount that Chloe brought. Olivia brought 2 times the amount that Luca brought. Together, all four students brought a total of 778 dollars to the mall. How múch money did each student bring to the mall?
a) Let:
- Let x represent the amount of money Luca brought.
- Let y represent the amount of money Chloe brought.
- Let z represent the amount of money Nico brought.
- Let w represent the amount of money Olivia brought.
b) From the information given:
- y = (1/2)x + 5
- z = 2y - 2
- w = 2x
- x + y + z + w = 778
Now we can substitute the expressions for y, z, and w into the last equation:
x + [(1/2)x + 5] + [2(1/2)x + 5 - 2] + 2x = 778
x + (1/2)x + 5 + x + 3 + 2x = 778
(7/2)x + 8 = 778
(7/2)x = 770
x = 220
Therefore, Luca brought $220 to the mall. Now we can find the amounts the other students brought.
y = (1/2)(220) + 5 = 110 + 5 = 115
z = 2(115) - 2 = 230 - 2 = 228
w = 2(220) = 440
c) Checking:
220 + 115 + 228 + 440 = 778
d) Therefore, Luca brought $220, Chloe brought $115, Nico brought $228, and Olivia brought $440 to the mall.