Problem 6 - Basic Engineering Mathematics

Question

Evaluate \(1 \frac{3}{5} \times 2 \frac{1}{3} \times 3 \frac{3}{7}\)

Step-by-Step Solution

Answer

The product is 12 \(\frac{4}{5}\).

Step 1: Convert Mixed Numbers to Improper Fractions

To evaluate the expression, first, we need to convert each mixed number into an improper fraction. For the first mixed number:\(1 \frac{3}{5} = \frac{5 \times 1 + 3}{5} = \frac{8}{5}\)For the second mixed number:\(2 \frac{1}{3} = \frac{3 \times 2 + 1}{3} = \frac{7}{3}\)For the third mixed number:\(3 \frac{3}{7} = \frac{7 \times 3 + 3}{7} = \frac{24}{7}\)

Step 2: Multiply the Improper Fractions

Next, multiply the improper fractions obtained in Step 1. So, we have:\(\frac{8}{5} \times \frac{7}{3} \times \frac{24}{7}\)First, multiply the first two fractions:\(\frac{8}{5} \times \frac{7}{3} = \frac{8 \times 7}{5 \times 3} = \frac{56}{15}\)Then, multiply the result by the third fraction:\(\frac{56}{15} \times \frac{24}{7} = \frac{56 \times 24}{15 \times 7} = \frac{1344}{105}\)

Step 3: Simplify the Resulting Fraction

Now, simplify the fraction \(\frac{1344}{105}\). Begin by finding the greatest common divisor (GCD) of 1344 and 105.The GCD of 1344 and 105 is 21.Divide both the numerator and the denominator by their GCD:\(\frac{1344 \div 21}{105 \div 21} = \frac{64}{5}\)

Step 4: Convert Fraction to Mixed Number

Finally, convert the improper fraction back into a mixed number.\(\frac{64}{5}\) can be expressed as:Divide 64 by 5, which results in 12 with a remainder of 4. So, \(\frac{64}{5} = 12 \frac{4}{5}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that consist of a whole number and a proper fraction. They are often used in everyday life when numbers need to be expressed more naturally, such as when measuring quantities like cups or pounds. A mixed number combines the wholes with the fractional parts.

For example, in the mixed number \(1 \frac{3}{5}\), "1" is the whole number, and "\(\frac{3}{5}\)" is the fractional part. It's beneficial to convert mixed numbers to improper fractions, especially when performing arithmetic operations like multiplication or addition.

To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction part.
• Place the sum over the original denominator.

For \(1 \frac{3}{5}\), you multiply 1 by 5 and add 3, resulting in 8. Therefore, \(1 \frac{3}{5} = \frac{8}{5}\). This simplification allows for easier calculation during multiplication of fractions.

Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. In simple terms, the top number is either larger than or the same as the bottom number. Improper fractions are the go-to format for executing fractional arithmetic more cleanly.

When multiplying fractions, it is easiest to convert mixed numbers to improper fractions first so there are no whole numbers involved in the multiplication.

For instance, using the conversion showed above, \(1 \frac{3}{5}\) becomes \(\frac{8}{5}\), simplifying the equation while multiplying. In the multiplication process, you:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

In our example: \(\frac{8}{5} \times \frac{7}{3} = \frac{56}{15}\). This straightforward process avoids confusion handling whole and fractional parts separately.

Simplification of Fractions

Simplification of fractions means reducing the fraction to its simplest form. This process makes the fraction easier to understand and use in subsequent operations. To simplify, you must find the greatest common divisor (GCD) of both the numerator and the denominator.

In the exercise, the resulting fraction from multiplication \(\frac{1344}{105}\) was simplified by finding the GCD, which was 21.

To simplify:

• Divide both the numerator and the denominator by the GCD.

Therefore, \(\frac{1344}{105} = \frac{64}{5}\) after division by 21.

Once simplified, if the fraction remains improper, it can be further converted back to a mixed number. This gives a more relatable sense of quantity, like the resulting \(\frac{64}{5} = 12 \frac{4}{5}\). Simplification is key to making mathematical solutions clear and concise, removing confusing larger numerals.