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Chapter 7: Problem 36

A city has a population of \(245,300\) which is \(110 \%\) of the population from the previous year. What was the population the previous year?

Short Answer

Expert verified
The population the previous year was 223,000.

Step by step solution

01

Understand the given values

The current population is given as 245,300, which is 110% of the previous year's population. This means that the current population is 110% of some value that we need to find.
02

Set up the equation

Let's denote the previous year's population as \(P\). According to the given information, 110% of \(P\) equals 245,300. We can write this as an equation: \(1.10P = 245,300\)
03

Solve for the previous year's population

To find the previous year's population \(P\), divide both sides of the equation by 1.10: \(P = \frac{245,300}{1.10}\)
04

Calculate the value

Perform the division to find \(P\): \(P = 223,000\)
05

Verify the calculation

Multiply 223,000 by 1.10 to check the solution: \(223,000 \times 1.10 = 245,300\). Since this matches the given current population, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

percentage increase
In everyday life, we often come across percentage increases. In this problem, we know the current population is 245,300, which is 110% of the previous year's population. So, what does it mean when we say 110%? This tells us that the current population includes the entire population from the previous year plus an additional 10%. To visualize:
  • 100% represents the previous year's population.
  • An additional 10% indicates the increase from last year to this year.
By understanding that 110% means 1.10 times the previous population, solving this kind of problem becomes straightforward.
algebraic equation
Algebra is a powerful tool to solve real-world problems by setting up equations. In this problem, the equation we set up is based on the percentage relationship. Let鈥檚 denote the previous year's population by the variable \(P\). According to the problem:
  • 110% of \(P\) equals the given current population, \(245,300\).
We can express this mathematically as \(1.10P = 245,300\). This equation states that 1.10 times the previous population equals the current population. Algebra allows us to isolate the variable \(P\) and find its value.
verifying solutions
Verifying solutions is a critical step to ensuring our calculations are correct. After calculating the previous population, it's wise to check if our solution holds true. Multiply the calculated previous year's population by 1.10 to see if it gives us the current population:
  • If calculated \(P = 223,000\), then:
  • \(223,000 \times 1.10 = 245,300\).
This checks out with the given data and confirms our solution is correct. Verification helps avoid mistakes and ensures our answer is reliable.
division in algebra
When solving algebraic equations, division is often used to isolate the variable. In our problem, we have the equation:
\(1.10P = 245,300\).
To solve for \(P\), divide both sides by 1.10:
\(P = \frac{245,300}{1.10}\)
Performing this division gives:
\

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Most popular questions from this chapter

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It is customary to leave a \(15-20 \%\) tip for the server in a restaurant. However, when you are at a restaurant in a social setting, you probably do not want to take out a pencil and piece of paper to figure out the tip. It is more socially acceptable to compute the tip mentally. Try this method. Step 1: First, if the bill is not a whole dollar amount, simplify the calculations by rounding the bill to the next-higher whole dollar. Step 2: Take \(10 \%\) of the bill. This is the same as taking one-tenth of the bill. Move the decimal point to the left 1 place. Step 3: If you want to leave a 20\% tip, double the value found in step 2. Step 4: If you want to leave a \(15 \%\) tip, first note that \(15 \%\) is \(5 \%+10 \% .\) Therefore, add one-half of the value found in step 2 to the number in step 2 . Estimate a \(20 \%\) tip on a bill of \(\$ 18.79\)

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