Word Problems and Data Analysis: GED Test Prep (page 4)
Many students struggle with word problems. In this article, you will learn how to solve word problems with confidence by translating the words into a mathematical equation. Because the GED Mathematics Exam focuses on real-life situations, it's especially important for you to know how to make the transition from sentences to a math problem.
Translating Words into Numbers
The most important skill needed for word problems is the ability to translate words into mathematical operations. This list provides some common examples of English phrases and their mathematical equivalents.
- Increase means add.
- Less than means subtract.
- Times or product means multiply.
- Times the sum means to multiply a number by a quantity.
- Two variables are sometimes used together.
- Inequality signs are used for at least and at most, as well as less than and more than.
A number increased by five = x + 5.
10 less than a number = x – 10.
Three times a number = 3x.
Five times the sum of a number and three = 5(x + 3).
A number y exceeds five times a number x by ten.
y = 5x + 10
The product of x and 6 is greater than 2.
x × 6 > 2
When 14 is added to a number x, the sum is less than 21.
x + 14 < 21
The sum of a number x and 4 is at least 9.
x + 4 ≥ 9
When seven is subtracted from a number x, the difference is at most 4.
x – 7 ≤ 4
Assigning Variables in Word Problems
It may be necessary to create and assign variables in a word problem. To do this, first identify an unknown and a known. You may not actually know the exact value of the "known," but you will know at least something about its value.
Max is three years older than Ricky.
Unknown = Ricky's age = x. Known = Max's age is three years older.
Therefore, Ricky's age = x and Max's age = x + 3.
Lisa made twice as many cookies as Rebecca.
Unknown = number of cookies Rebecca made = x.
Known = number of cookies Lisa made = 2x.
Cordelia has five more than three times the number of books that Becky has.
Unknown = the number of books Becky has = x.
Known = the number of books Cordelia has = 3x + 5.
A ratio is a comparison of two quantities measured in the same units. It can be symbolized by the use of a colon—x : y or or x to y. Ratio problems can be solved using the concept of multiples.
A bag containing some red and some green candies has a total of 60 candies in it. The ratio of the number of green to red candies is 7:8.How many of each color are there in the bag?
From the problem, it is known that 7 and 8 share a multiple and that the sum of their product is 60. Therefore, you can write and solve the following equation:
7x + 8x = 60
15x = 60
Therefore, there are 7x = (7)(4) = 28 green candies and 8x = (8)(4) = 32 red candies.
Mean, Median, and Mode
To find the average, or mean, of a set of numbers, add all of the numbers together and divide by the quantity of numbers in the set.
Find the average of 9, 4, 7, 6, and 4.
(Divide by 5 because there are 5 numbers in the set.)
To find the median of a set of numbers, arrange the numbers in ascending order and find the middle value.
- If the set contains an odd number of elements, then simply choose the middle value.
- If the set contains an even number of elements, simply average the two middle values.
Find the median of the number set: 1, 3, 5, 7, 2.
First arrange the set in ascending order: 1, 2, 3, 5, 7,and then choose the middle value: 3. The answer is 3.
Find the median of the number set: 1, 5, 3, 7, 2, 8.
First arrange the set in ascending order: 1, 2, 3, 5, 7, 8, and then choose the middle values, 3 and 5.
Find the average of the numbers 3 and 5:
= 4. The median is 4.
The mode of a set of numbers is the number that occurs the greatest number of times.
For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs the most often.
A percent is a measure of a part to a whole, with the whole being equal to 100.
- To change a decimal to a percentage, move the decimal point two units to the right and add a percentage symbol.
- To change a fraction to a percentage, first change the fraction to a decimal. To do this, divide the numerator by the denominator. Then change the decimal to a percentage.
- To change a percentage to a decimal, simply move the decimal point two places to the left and eliminate the percentage symbol.
- To change a percentage to a fraction, put the percent over 100 and reduce.
.45 = 45%
.07 = 7%
.9 = 90%
64% = .64
87% = .87
7% = .07
Keep in mind that any percentage that is 100 or greater will need to reflect a whole number or mixed number when converted.
Here are some conversions you should be familiar with. The order is from most common to less common.
Interest is a fee paid for the use of someone else's money. If you put money in a savings account, you receive interest from the bank. If you take out a loan, you pay interest to the lender. The amount of money you invest or borrow is called the principal. The amount you repay is the amount of the principal plus the interest. The formula for simple interest is found on the formula sheet in the GED. Simple interest is a percent of the principal multiplied by the length of the loan:
- Interest = principal × rate × time
Sometimes it may be easier to use the letters of each as variables:
- I = prt
Michelle borrows $2,500 from her uncle for three years at 6% simple interest. How much interest will she pay on the loan?
Michelle will pay $450 in interest.
Some problems will ask you to find the amount that will be paid back from a loan. This adds an additional step to problems of interest. In the previous example, Michelle will owe $450 in interest at the end of three years. However, it is important to remember that she will pay back the $450 in interest as well as the principal, $2,500. Therefore, she will pay her uncle $2,500 + $450 = $2,950.
In a simple interest problem, the rate is an annual, or yearly, rate. Therefore, the time must also be expressed in years.
Kai invests $4,000 for nine months. Her investment will pay 8%. How much money will she have at the end of nine months?
Kai will earn $180 in interest.
Probability is expressed as a fraction and measures the likelihood that a specific event will occur. To find the probability of a specific outcome, use this formula:
If a bag contains 5 blue marbles, 3 red marbles, and 6 green marbles, find the probability of selecting a red marble:
Helpful Hints about Probability
- If an event is certain to occur, the probability is 1.
- If an event is certain not to occur (impossible), the probability is 0.
- If you know the probability of all other events occurring, you can find the probability of the remaining event by adding the known probabilities together and subtracting their total from 1.
Graphs and Tables
The GED Mathematics Exam will test your ability to analyze graphs and tables. It is important to read each graph or table very carefully before reading the question. This will help you to process the information that is presented. It is extremely important to read all of the information presented, paying special attention to headings and units of measure. Here is an overview of the types of graphs you will encounter:
- Circle graphs or pie charts
- Bar graphs
- Broken line graphs
This type of graph is representative of a whole and is usually divided into percentages. Each section of the chart represents a portion of the whole, and all of these sections added together will equal 100% of the whole.
Bar graphs compare similar things with different length bars representing different values. Be sure to read all labels and legends, looking carefully at the base and sides of the graph to see what the bars are measuring and how much they are increasing or decreasing.
Broken line graphs illustrate a measurable change over time. If a line is slanted up, it represents an increase whereas a line sloping down represents a decrease. A flat line indicates no change as time elapses.
Scientific notation is a method used by scientists to convert very large or very small numbers to more manageable ones. You will have to make a few conversions to scientific notation on the GED Mathematics Exam. Expressing answers in scientific notation involves moving the decimal point and multiplying by a power of ten.
A space satellite travels 46,000,000 miles from the earth. What is the number in scientific notation?
Step 1: Starting at the decimal point to the right of the last zero, move the decimal point until only one digit remains to its left:
46,000,000 becomes 4.6.
Step 2: Count the number of places the decimal was moved left. In this example, the decimal point was moved 7 places, and express it as a power of 10: 107
Step 3: Express the full answer in scientific notation by multiplying the reduced answer from step 1 by 107:
4.6 × 107
An amoeba is .000056 inch long. What is its length in scientific notation?
Step 1: Move the decimal point to the right until there is one digit other than zero to the left of the decimal.
.000056 becomes 5.6
Step 2: Count the number of places moved to the right—5.However, because the value of the number is being increased as it is expressed in scientific notation, it is written as a negative exponent. 10–5
Step 3: Express the full answer in scientific notation: .0000056 becomes 5.6 × 10–5
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