Question

(a) Compute the value of (1) \(90 \%\) of 400 (2) \(180 \%\) of 400 (3) \(50 \%\) of 500 (4) \(200 \%\) of 4 (b) What percent of (1) 100 is \(99.5 ?\) (2) 200 is \(4 ?\)

Short answer

(a) (1) \(90 \%\) of 400 is 360. (2) \(180 \%\) of 400 is 720. (3) \(50 \%\) of 500 is 250. (4) \(200 \%\) of 4 is 8. (b) (1) \(99.5\) is \(99.5 \%\) of 100. (2) \(4\) is \(2 \%\) of 200.

Step-by-step solution

Compute \(90 \%\) of 400

We need to find the \(90 \%\) of 400. The formula for this calculation is: Percentage value = (Percentage/100) * Original number Plugging in the values, we get: Percentage value = (90/100) * 400 = 0.9 * 400 = 360

Compute \(180 \%\) of 400

We need to find the \(180 \%\) of 400. Plugging in the values into our formula, we get: Percentage value = (180/100) * 400 = 1.8 * 400 = 720

Compute \(50 \%\) of 500

We need to find the \(50 \%\) of 500. Plugging in the values into our formula, we get: Percentage value = (50/100) * 500 = 0.5 * 500 = 250

Compute \(200 \%\) of 4

We need to find the \(200 \%\) of 4. Plugging in the values into our formula, we get: Percentage value = (200/100) * 4 = 2 * 4 = 8 For part (b):

Find the percentage of 100 that is 99.5

We need to find what percent 99.5 is of 100. The formula we will use is: Percentage = (Percentage value / Original number) * 100 Plugging in the values, we get: Percentage = (99.5 / 100) * 100 = 99.5\%

Find the percentage of 200 that is 4

We need to find what percent 4 is of 200. Plugging in the values into our formula, we get: Percentage = (4 / 200) * 100 = 2 \%

Key concepts

Percent Increase

Understanding percentage increase is a fundamental skill in both everyday life and academics. When we talk about percentage increase, we refer to how much a quantity has gone up in terms of percentage relative to its original value. To calculate the percentage increase, we use the formula:\[ \text{Percent Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100 \]This formula helps us to see the increase as a proportion of the initial amount, converted into a percentage.
For example, if you had 400 apples and they increased to 500, the increase would be:

• New Value: 500
• Original Value: 400
• Increase: 500 - 400 = 100
• Percent Increase: \(\left( \frac{100}{400} \right) \times 100 = 25\% \)

Practicing these calculations helps to easily spot and understand changes in data.

Basic Arithmetic

Basic arithmetic is essential for performing percentage calculations efficiently. Arithmetic primarily involves operations like addition, subtraction, multiplication, and division.
Understanding multiplication and division is particularly crucial in percentage calculations. Multiplication is used when finding a certain percentage of a number, as seen in exercises where you calculate values like 90% or 50% of given quantities.

• To find 90% of 400:
  Multiply 400 by 0.9, which comes from converting 90% to a decimal: \(400 \times 0.9 = 360\).
• Division is used when converting a percentage back to a number, which is done by using the formula for percentages: \((\text{Decimal} \times 100)\) to return to its percentage form.

Grasping these arithmetic operations simplifies many mathematical processes, making complex problems easier to manage.

Mathematical Formulas

Mathematical formulas are the backbone of solving percentage problems methodically. Using the right formula makes the calculations straightforward and reduces the likelihood of error. Here are the primary formulas used in this exercise:

• To calculate a percentage value from a number: \( \text{Value} = \left( \frac{\text{Percentage}}{100} \right) \times \text{Original Number} \)
• For example, to find 180% of 400, use: \( \left( \frac{180}{100} \right) \times 400 = 720 \).
• To find what percentage one number is of another: \( \text{Percentage} = \left( \frac{\text{Value}}{\text{Original Number}} \right) \times 100 \)
• For example, what percent of 100 is 99.5? Use: \( \left( \frac{99.5}{100} \right) \times 100 = 99.5\% \).

Remembering these formulas allows you to tackle any percentage-related problem with ease.

Step-by-Step Solutions

Applying step-by-step solutions in percentage problems ensures accuracy and clarity when solving complex calculations. Breaking down problems into smaller steps is key to understanding and solving them effectively.
For instance, let's analyze the calculation of "90% of 400":

• Step 1: Convert the percentage to a decimal. 90% becomes 0.9.
• Step 2: Multiply the original value by this decimal. Calculate: \(400 \times 0.9 = 360\).
• Step 3: Confirm the solution by checking through the calculations once more to avoid errors.

By following a methodical approach, such as outlined above, students can gain confidence in tackling more complex problems and improve their problem-solving skills.