Question

Estimate each result using benchmarks. Then find the exact value. \(60 \%\) of 90

Short answer

60% of 90 is 54. The estimate using benchmarks was slightly above 45.

Step-by-step solution

Understand the Problem

We need to estimate 60% of 90 using benchmarks, and then find the exact value.

Estimate Using Benchmarks

Using benchmarks, we know that 50% of 90 is 45. Since 60% is slightly more than 50%, we estimate 60% of 90 to be a bit more than 45.

Calculate the Exact Value

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Key concepts

Estimating Percentages

Estimating percentages can be very useful when you need a quick and rough answer. This technique helps simplify complicated calculations, making it easier to understand and solve problems quickly.
Start by using familiar percentages that you know well, like 50%, 25%, or 10%. These are called benchmarks.
For example, if you need to estimate 60% of 90, you can start with 50%, which is a familiar benchmark.
Since 50% of 90 is 45, and 60% is just a bit more than 50%, you can estimate that 60% of 90 is slightly more than 45. This quick estimation helps you to easily check the reasonableness of your answer before doing the exact calculation.

Benchmarks in Mathematics

Benchmarks in mathematics are reference points that help us make quick and reasonable estimates. These reference points are usually easy-to-remember values, such as:

• 50% (half of a number)
• 25% (a quarter of a number)
• 10% (a tenth of a number)

Using benchmarks makes it easier to handle percentages in everyday scenarios.
For instance, if you want to estimate 70% of 200, you can break it down using familiar benchmarks:

• 50% of 200 is 100
• 10% of 200 is 20
• Adding these: 100 + 20 + 20 (since 70% is 50% + 10% + 10%) results in 140

This method allows you to quickly estimate the answer without extensive calculations. It’s also useful for double-checking your results.

Exact Value Calculation

After estimating, it's important to calculate the exact value to verify the accuracy of your estimate. Calculating the exact value of a percentage involves a more precise method.
To find 60% of 90, follow these steps:

• Convert the percentage to a decimal: 60% becomes 0.60
• Multiply the decimal by the number: 0.60 × 90
• This equals 54, which means 60% of 90 is exactly 54

While estimation helps you understand the scale of your answer, exact calculation ensures you have the precise figure needed. Combining both approaches—estimation for a quick check and exact calculation for accuracy—gives you a solid foundation for solving percentage problems.