Question 3Simplify The Following Without Using A Calculator:(a) ${ 2 \frac{1}{4} \times \frac{2}{3}\$} (b) { \frac{3}{4} + 4 \frac{1}{6} \div 5$}$Tarra Bought A Fridge For R2 500 And Sold It For R3 999.Calculate The Percentage Profit

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Introduction

In this article, we will simplify two mathematical expressions without using a calculator and calculate the percentage profit on the sale of a fridge. We will use basic arithmetic operations such as multiplication, division, addition, and subtraction to simplify the expressions.

Simplifying Expression (a)

Multiplying Mixed Numbers

To simplify the expression ${2 \frac{1}{4} \times \frac{2}{3}\$}, we need to multiply the mixed numbers. We can start by converting the mixed numbers to improper fractions.

Converting Mixed Numbers to Improper Fractions

A mixed number is a combination of a whole number and a fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

  • ${$2 \frac{1}{4}$ can be converted to an improper fraction as follows:

    [$2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$]

  • [$\frac{2}{3}$ is already an improper fraction.

Multiplying Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can multiply them.

Multiplying Improper Fractions

To multiply improper fractions, we multiply the numerators and multiply the denominators.

  • [$\frac{9}{4} \times \frac{2}{3} = \frac{(9 \times 2)}{(4 \times 3)} = \frac{18}{12}$]

Simplifying the Result

The result of the multiplication is [$\frac{18}{12}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Finding the Greatest Common Divisor (GCD)

The GCD of 18 and 12 is 6.

  • [$\frac{18}{12} = \frac{(18 \div 6)}{(12 \div 6)} = \frac{3}{2}$]

Therefore, the simplified expression is [$\frac{3}{2}$.

Simplifying Expression (b)

Adding and Subtracting Fractions

To simplify the expression [\frac{3}{4} + 4 \frac{1}{6} \div 5\$}, we need to add and subtract fractions. We can start by converting the mixed number to an improper fraction.

Converting Mixed Numbers to Improper Fractions

A mixed number is a combination of a whole number and a fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

  • ${$4 \frac{1}{6}$ can be converted to an improper fraction as follows:

    [$4 \frac{1}{6} = \frac{(4 \times 6) + 1}{6} = \frac{24 + 1}{6} = \frac{25}{6}$]

Dividing by a Whole Number

To divide by a whole number, we can multiply by the reciprocal of the whole number.

  • [$\frac{25}{6} \div 5 = \frac{25}{6} \times \frac{1}{5} = \frac{25}{30}$]

Adding Fractions

Now that we have converted the mixed number to an improper fraction and divided by a whole number, we can add the fractions.

Adding Fractions

To add fractions, we need to have the same denominator. We can find the least common multiple (LCM) of the denominators to add the fractions.

  • The LCM of 4 and 30 is 60.

  • [$\frac{3}{4} = \frac{(3 \times 15)}{(4 \times 15)} = \frac{45}{60}$

  • [$\frac{25}{30} = \frac{(25 \times 2)}{(30 \times 2)} = \frac{50}{60}$

Adding the Fractions

Now that we have the same denominator, we can add the fractions.

  • [$\frac{45}{60} + \frac{50}{60} = \frac{45 + 50}{60} = \frac{95}{60}$]

Therefore, the simplified expression is [$\frac{95}{60}$.

Calculating the Percentage Profit

Finding the Profit

To calculate the percentage profit, we need to find the profit. The profit is the difference between the selling price and the cost price.

  • The selling price is R3 999.

  • The cost price is R2 500.

  • [$\text{Profit} = \text{Selling Price} - \text{Cost Price}$

  • [\text{Profit} = 3999 - 2500 = 1499\$}

Finding the Percentage Profit

To find the percentage profit, we can use the formula:

  • {\text{Percentage Profit} = \frac{\text{Profit}}{\text{Cost Price}} \times 100$}$

  • [$\text{Percentage Profit} = \frac{1499}{2500} \times 100 = 59.96%$]

Therefore, the percentage profit is 59.96%.

Conclusion

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. For example, [$2 \frac{1}{4}$ can be converted to an improper fraction as follows:

  • [$2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$]

Q: How do I multiply improper fractions?

A: To multiply improper fractions, you multiply the numerators and multiply the denominators. For example, [$\frac{9}{4} \times \frac{2}{3} = \frac{(9 \times 2)}{(4 \times 3)} = \frac{18}{12}$]

Q: How do I simplify a fraction?

A: To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). For example, [$\frac{18}{12} = \frac{(18 \div 6)}{(12 \div 6)} = \frac{3}{2}$]

Q: How do I add and subtract fractions?

A: To add and subtract fractions, you need to have the same denominator. You can find the least common multiple (LCM) of the denominators to add the fractions. For example, [$\frac{3}{4} + \frac{25}{6} = \frac{45}{60} + \frac{50}{60} = \frac{95}{60}$]

Q: How do I calculate the percentage profit?

A: To calculate the percentage profit, you use the formula: [$\text{Percentage Profit} = \frac{\text{Profit}}{\text{Cost Price}} \times 100$. For example, if the profit is R1 499 and the cost price is R2 500, the percentage profit is:

  • [$\text{Percentage Profit} = \frac{1499}{2500} \times 100 = 59.96%$]

Q: What is the difference between a percentage profit and a percentage loss?

A: A percentage profit is the percentage increase in value, while a percentage loss is the percentage decrease in value.

Q: How do I calculate the percentage loss?

A: To calculate the percentage loss, you use the formula: [$\text{Percentage Loss} = \frac{\text{Loss}}{\text{Cost Price}} \times 100$. For example, if the loss is R1 499 and the cost price is R2 500, the percentage loss is:

  • [$\text{Percentage Loss} = \frac{1499}{2500} \times 100 = 59.96%$]

Q: What is the importance of calculating the percentage profit or loss?

A: Calculating the percentage profit or loss is important because it helps you understand the financial performance of a business or investment. It also helps you make informed decisions about whether to invest or divest in a particular venture.

Q: How do I apply the concepts of percentage profit and loss in real-life scenarios?

A: You can apply the concepts of percentage profit and loss in real-life scenarios such as:

  • Calculating the profit or loss on a sale or purchase
  • Evaluating the financial performance of a business or investment
  • Making informed decisions about whether to invest or divest in a particular venture
  • Understanding the impact of inflation or deflation on the value of an investment

By understanding the concepts of percentage profit and loss, you can make informed decisions and achieve your financial goals.