Algebra is about expressing a relationship between two changing quantities. For example, as the height of a person changes, so does the weight of the person. Also, when you earn money, you may have a pay rate that tells you how much money you will have after working for an amount of time. Algebra helps you write down that relationship without using words. You can then use the equation to answer questions about specific time points or conditions.
For word problems, express what information you have. Express the information you want to know as a variable. Pay attention to language.
There are many words used for addition: plus, sum, add, combine, all together, increase, join, total, both, how many in all?
Words used for subtraction: minus, less than, decrease, deduct, difference, take away, subtract, more than, reduce, remain, left, change, comparisons.
Words that imply multiplication: times, multiplied, (percent) of, twice, doubled (2 times), tripled (3 times), apiece, area, volume, product.
Words that imply division: split, each, shared, equal pieces, out of, per, every, average, part, ratio, quotient (the answer to a division problem).
Knowing what each word implies helps you identify the relationships between variables.
Example 1:
You want to make a bird house from a 6 foot long board. If you need 4 pieces of wood the same length plus one piece 4 inches longer, how long can the 4 similar pieces be?
Define your variable as the length of one of the similar boards, s.
The length of the longer board would be s + 4.
Convert 6 feet to inches: 6 feet x 12 inches per foot = 72 inches.
Define the relationship: Add up the length of the boards: 4s + (s + 4) = 72
We can then simplify and solve the equation: 5s + 4 = 72.
Subtract 4 from each side of the equation: 5 s = 72 - 4, so s = 68 / 5 or s equals 13.6 inches
Always check the units of your answer! If we had just used the metric system to measure, our decimal answer would make more sense! Units will help you to measure correctly 0.6 inches is 6 out of 10, but there are 8 or 16 equal divisions marked on a ruler for inches!
That gives us a new ratio problem: 0.6 can be expressed as 6 / 10 = x / 16
So to find out how many sixteenths (x) we need, we multiply both sides by 16: 16 * (6 / 10) = x or x = 9.6 sixteenths.
How does that help us, we might ask? Well, if you cut the wood to 13 and 10 sixteenths inches (13 5/8"), you have made an estimate, but it is a lot closer than the mistaken measurement of 13 and 6 /8 (3/4"), which is too big or 13 and 6/16 (3/8"), which is too small.
Back to our bird house...we can now cut 4 pieces of wood 13 and 9.6 / 16 inches long and one piece 17 and 9.6 / 16 inches long.
We can check our answer by adding 13 + 13 + 13 + 13 + 17 + (5* 9.6)/16 = 52 + 17 + 3 = 72 inches!
Money can be earned at a certain rate (dollars per hour), but distance problems also involve rates of travel and are sometimes more confusing. Keeping track of units can help you solve the problem, since units have to cancel to get the right answer. Distance is measured in meters and speed is given in meters per second (or kilometers per hour), so speed times amount of time equals distance
90 km / hour * number of hours = distance traveled (in km)
25 meter/s * number of seconds = distance traveled (in meters)
Example 2:
If the car you are riding in travels at 90 km per hour, how long will it take you to travel the 139 miles from Washington, D.C. to Philadelphia, Pennsylvania?
Define your variable as the length of time in hours to get to Philadelphia, h.
From the formula above (rate * time = distance), we cam rearrange the equation to show that 139 miles divided by the rate equals the time it takes.
Convert 139 miles to kilometers: 139 miles * 1.61 kilometers / mile = 224 km (rounded up).
Define the relationship: 224 km / (90 km/hour) = h (Notice how the units of kilometers cancel - one on the top, divided by one on the bottom)
We can then simplify and solve the equation. In this case, we rearranged the equation before we put the numbers into it, so we can just calculate the answer. 224 / 90 = 2.49 hours
Always check the units of your answer!
0.49 hours is close to half an hour (30 minutes). Let's convert that to minutes: 0.49 hours * 60 minutes / hour = 29.4 minutes For the whole trip, that's 2 hours 29 minutes and 24 seconds or about 2 1/2 hours.
Don't forget to check your answer. Is it reasonable? 90 km/hour is about 55 miles per hour. At 60 miles per hour, we would travel at one mile per minute, which would be 139 minutes ( 2 hours and 19 minutes). Since we are traveling a bit slower, our answer of 2 hours 29.4 minutes makes sense.
Let's do one more rate problem. We already know that distance = rate * time.
Example 3:
There are at least two ways to get to school in the morning. Carla likes to ride her bike to school when the weather is nice, but it takes her 45 minutes. If she rides the bus, the trip takes 20 minutes. If the bus travels 20 miles per hour faster than Carla biking, can you find out how far it is from Carla's house to the school?
Define your variable as the distance in miles from Carla's house to the school, d.
We know that distance = rate * time. Since Carla bikes the same distance as the bus travels, we can use the difference in rates of travel (20 mph) to determine the exact rate of travel of either Carla or the bus, using the formula Rate1 * Time1 = Rate2 * Time2. We can do this because the distance traveled is the same.
Write and expression to compare Carla's rate of travel with the rate of travel of the bus. We are going to give Carla's speed another variable name, s. If Carla travels at s mph, then the bus travels at (s + 20) mph.
Define the relationship: Carla travels 45 minutes * s mph. We forgot to convert minutes to hours!
Carla travels for 45 minutes * 1 hour/60 minutes = 0.75 hours; The bus travels for 20 minutes * 1 hour/ 60 minutes = 0.33 hours.
Carla travels 0.75 h * s mph = 0.33 h * (s + 20) mph for the bus.
We distribute the 0.33 hours by multiplying it by both items in parenthesis:
0.75 hours * s = (0.33 hours * s) + (0.33 hours * 20 mph)
We can then simplify and solve the equation.
0.75 s = 0.33 s + 6.6 mph
0.75s - 0.33s = 6.6 mph, so 0.42 s = 6.6 mph. If you divide by 0.42, you get
s = 15.7 mph, the rate of speed Carla travels. But we want to know distance.
Distance = Carla's rate * 0.75 hours = 15.7 mph * 0.75 hours = 11.8 miles. That's a long way to ride a bike and Carla is pretty fast!
Always check the units of your answer! We used hours and miles throughout our problem.
Don't forget to check your answer. Let's use the speed the bus is traveling (15. 7 mph + 20 mph) to check our answer.
Bus travels 35.7 mph * 0.33 hours = 11.8 miles. I guess the bus is pretty speedy as well.
Whenever you get a word problem, first write what you know.
Define the variables. Convert units. Relate the facts. Simplify and Solve by using the same operation on both sides of the equation. Distribute values across the parentheses, when there are parentheses. Check your units. Check that you answered the right question. Check that your answer makes sense.