Algebra Percents and Simple Interest Study Guide (page 2)
Introduction to Algebra Percents and Simple Interest
Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.
In this lesson, you'll learn how to use algebra to find the percent increase or percent decrease between two values and how to calculate simple interest.
A percent is a number out of one hundred. For example, 36% is 36 out of 100. We can answer some simple percent questions without algebra:
What is 10% of 60? 10% = 0.10. Multiply 60 by 0.10: (60)(0.10) = 6. Therefore, 6 is 10% of 60.
What percent is 12 of 48? Divide 12 by 48: 12 ÷ 48 = 0.25, 0.25 = 25%. Therefore, 12 is 25% of 48.
Finally, 20% of what number is 15? 20% = 0.20. Divide 15 by 0.20: 15 ÷ 0.20 = 75. Therefore, 15 is 20% of 75.
When percent questions get a bit tougher, we can use algebra to help us find the answers. If a value increases from 12 to 15, we can find the percent increase by subtracting the original value from the new value and dividing by the original value:
But what if we were given the original value and the percent increase, and we needed to find the new value? We can use the same formula, but let x represent the new value.
What is 20 after a 40% increase?
The original value is 20 and the new value is x. Because the percent increase is equal to the new value minus the old value divided by the old value, subtract 20 from x and divide by 20. Set that fraction equal to 40%, which is 0.40:
Multiply both sides of the equation by 20, and then add 20 to both sides:
- x – 20 = 8
- x = 28
Algebra also comes in handy if we are given a new value after a percent increase and we are asked to find the original value.
After a 24% increase, a value is now 80.6. What was the original value?
We can use the same formula that we used to solve the preceding question, but this time, we let x represent the original value instead of the new value. The difference between the new value and the original value is 80.6 – x. We divide that by the original value, x, and set it equal to the percent increase, 24%, or 0.24:
Multiply both sides of the equation by x, and then add x to both sides:
0.24x = 80.6 – x
1.24x = 80.6
Divide both sides by 1.24:
x = 65
The formula for percent decrease is very similar to the one for percent increase. If a value decreases from 15 to 12, we can find the percent decrease by subtracting the new value from the original value and dividing by the original value:
When finding a percent increase or decrease, be sure to divide the difference between the two values by the original value. The percent increase from one value to a second value is not equal to the percent decrease from that second value to the first value. Looking back at our two examples, going from 12 to 15 was a 25% increase, but going from 15 to 12 was only a 20% decrease. In both examples, the difference between 12 and 15 is 3, but in the percent increase example, we divided 3 by 12, because 12 was the original value, and in the percent decrease example, we divided 3 by 15, because 15 was the original value.
After a 68% decrease, a value is now 74. What was the original value?
Let x represent the original value. The difference between the original value and the new value is x – 74. Divide that by the original value, x, and set it equal to the percent decrease, 68%, which is 0.68:
Multiply both sides of the equation by x, and then subtract x from both sides:
0.68x = x – 74
–0.32x = –74
Divide both sides by –0.32:
x = 231.25
We can find how much interest an amount of money, or principal, has gained by multiplying the principal by an interest rate and a length of time. The formula for interest is I = prt. Interest, principal, rate, and time are all variables, and given any three of them, we can substitute those values into the equation to find the missing fourth value.
If a principal of $500 gains $60 in interest in three years, what was the interest rate per year?
The interest rate is the percent of the principal that is added as interest each year. Because I = prt and we are looking for the rate, r, we can divide both sides of the equation by pt to get r alone on the right side:
To find the rate, divide the interest by the product of the principal and the time: 60 ÷ (500)(3) = 60 ÷ 1,500 = 0.04 = 4%. The principal gained interest at a rate of 4% per year.
When calculating interest, be sure the interest rate and the time have consistent units of measure. If the interest rate is given on a yearly basis, the time must also be in years. If the interest rate is given on a yearly basis and the time is given in months, convert the time to years before using the interest formula.
If $36 in interest is gained over six months at a rate of 6% per year, how much was the principal?
We can rewrite the formula I = prt to solve for p by dividing both sides of the equation by rt: . The interest rate is given per year, but the length of time is given in months. Divide the number of months by 12, because there are 12 months in a year: 6 ÷ 12 = 0.5. The time is 0.5 years and the rate is 6%, or 0.06. Because $1,200. The principal was $1,200.
Find practice problems and solutions for these concepts at Algebra Percents and Simple Interest Practice Questions.
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