1. A summer camp counsellor wants to find a length, x, in feet, across the lake as represented in the sketch below. The lengths represented by AB, EB, BD and CD on the sketch were determined to be 1800 feet, 1400 feet, 700 feet, and 800 feet, respectively. Segments AC and DE intersect at B, and angles AEB and CDB have the same measure. What is the value of x?

Solution:

Since the angles AEB and BDC have the same measure as well as angles ABE and CBD, triangles (two pairs of congruent corresponding angles). Since the triangles EAB DCB are similar, their corresponding sides are in the same proportion; thus CD/x = BD/EB. Substituting the given values, then CD/x = BD/EB becomes 800/x = 700/1400.

Therefore, x = (800 * 1400) / 700 = 1600 feet.

2. A food truck sells salads for $6.50 each and drinks for $2.00 each. The food truck’s revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that day?

Solution:

Let x = total number of salads sold and y = total number of drinks sold.

x + y = 209, y = 209 – x, and
6.50x + 2.00y = 836.50, 6.5x + 2(209 – x) = 836.5,  4.5x + 418 = 836.5 then, 
4.5 x = 418.5 and
x = 93 salads.

3. How many litres of a 25% saline solution must be added to 3 litres of a 10% saline solution to obtain a 15% saline solution?

Solution:

Let x = the amount, in litres, of a 25% saline solution to be added. 

Then 3(0.10) + 0.25x = 0.15(x + 3), then 0.25x – 0.15x = 0.45 – 0.30 
Therefore, 0.10x = 0.15 or
x = 1.5 litres.