Second angle of a triangle is 20° more than the first. The measure of the third angle is twice the measure of the first angle. Find all three angles.
I invest $30,000 for one year. Part is invested at 2% interest per annum and the rest is invested at 3% per annum. I earn $800 after one year. How much did I invest at 2% and how much did I invest at 3%?
General Method to Solve First, read the problem carefully. Decide what information you are given and what is being asked. Draw a diagram if applicable. Remember, there is more than one piece of information missing in all these problems. After you have done this, do the following:
Define only one variable. Let n = (describe in English what n equals)
n = 40
Step 5: second angle = n + 20 = 40 + 20 = 60
third angle = 2n = 2(40) = 80
Answer: 40°, 60°, 80° Problem 3 – invest $30,000 for one year, part at 2%, part at 3%, earn $800 Step 1: Let x = amount invested at 2%
Step 2: $30,000 – x = amount invested at 3%
Step 3: Remember I = prt. Here, interest (I) ($800) will equal interest on amount (x) at 2% (.02) and interest on amount ($30,000 –x ) at 3% (.03). t is one year. Hence, x(.02)(1) + (30,000 – x)(.03)(1) = 800
-.01x = -100 (subtracting 800 from both sides of equation)
x = $10,000 (dividing both sides by .01)
x = $10,000 = amount invested at 2%
$30,000 – x = $30,000 - $10,000 = $20,000 = amount invested at 3%
New Types of Problems
Solve other types of problems the same way.
Distance = Rate x Time Problems Example 1: Two trains leave Los Angeles at the same time. Train A travels north. Train B travels south. At the end of two hours they are 180 miles apart. Find the rate of both trains if Train A is traveling 10 miles per hour slower than Train B.
Preliminary Steps: On these types of problems it is helpful to draw a diagram and a chart. Also, remember that distance = rate x time. Rate is the same as speed. The other steps are the same.
Step 3: The total distance, 180 miles, equals the distance train A went plus the distance train B went. Distance = rate multiplied by time or D = rt. t = 2 hours
2x + 2(x + 10) = 180
2x + 2(x + 10) = 180
2x + 2x + 20 = 180 (distributive property)
4x + 20 = 180 (combine like terms)
4x = 160 (subtract 20 from both sides)
x = 40 (Divide both sides by 4)
Step 5: Train A’s rate was 40 mph. Train B’s rate was x + 10 = 50 mph.
1 It is good form to include the units on all calculations and perform the dimensional analysis. This also helps insure you are doing the problem correctly. For sake of simplicity here, however, I have not included the units or dimensional analysis.