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Chapter 4: Problem 8

Estimate each result using benchmarks. Then find the exact value. \(75 \%\) of 80

Short Answer

Expert verified
Estimate: 60; Exact value: 60

Step by step solution

01

Identify the Benchmark

A helpful benchmark for this problem is 75%. Since 75% is close to 3/4 (three-quarters), use this fraction to estimate the result.
02

Estimate Using the Benchmark

Calculate 3/4 of 80 as an estimate. Step1: Divide 80 by 4: \[ 80 \div\ 4 = 20 \]. Step 2: Then multiply the result by 3: \[ 20 \times\ 3 = 60 \]. Thus the estimate is 60.
03

Convert the Percentage to a Decimal

To find the exact value, first convert 75% to a decimal: 75% = 0.75.
04

Multiply to Find the Exact Value

Now, multiply 80 by 0.75: \[ 80 \times\ 0.75 = 60 \]. So, the exact value is 60.

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Key Concepts

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

estimating percentages
Estimating percentages involves finding an approximate value that is close to the actual percentage calculation. It can be useful for quickly checking your work or making quick decisions in everyday situations. For example, in our exercise, we estimated 75% of 80 by using benchmark percentages, which simplifies the calculation process. This practice not only helps in making fast mental calculations but also aids in understanding how percentages work. To estimate percentages effectively, look for whole numbers or simple fractions close to the given percentage.
benchmark percentages
Benchmark percentages are standard percentages that can be easily calculated and used for estimating other percentages. Common benchmarks include 25% (or 1/4), 50% (or 1/2), and 75% (or 3/4). In our example, 75% of 80 can be estimated by using the benchmark 3/4. This makes the calculation straightforward:
1. Find 1/4 of 80 (which is easy because you are dividing by 4)
2. Multiply the result by 3 to get the value for 3/4.
This technique simplifies complex percentage calculations into simpler multiplication or division tasks.
converting percentages to decimals
Converting percentages to decimals is a crucial step in many percentage calculations. To convert a percentage to a decimal, you simply divide by 100. This is because 'percent' means 'per 100.'
For example, 75% becomes 0.75 by dividing 75 by 100. This conversion is necessary when performing mathematical operations, like multiplication, because it simplifies the calculation process. Knowing how to convert between percentages and decimals will help with tasks like finding exact values.
multiplying decimals
Multiplying decimals is an essential arithmetic skill when dealing with percentages in decimal form. After converting the percentage to a decimal, you multiply it by the given number to find the exact value. Using our exercise:
1. Convert 75% to 0.75.
2. Multiply 0.75 by 80: \[ 80 \times 0.75 = 60 \]
The result is the exact value of 75% of 80. Practice multiplying decimals to get comfortable with handling percentages in various calculations.

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