Question

The average of \(4 \frac{3}{5}, 2 \frac{2}{3}, 6 \frac{8}{9}, 7 \frac{7}{15}, 3 \frac{5}{9}\) is (a) \(5 \frac{3}{225}\) (b) \(5 \frac{8}{225}\) (c) \(6 \frac{3}{45}\) (d) \(25 \frac{8}{45}\)

Short answer

\) (a) \(5 \frac{3}{5}\) (b) \(5 \frac{8}{225}\) (c) \(6 \frac{15}{225}\) (d) \(4 \frac{29}{225}\) Answer: (b) \(5 \frac{8}{225}\)

Step-by-step solution

Convert mixed numbers into improper fractions

To change a mixed number into an improper fraction, we need to use the following formula: \(\frac{(whole \thinspace number \times denominator) + numerator}{denominator}.\) For the given mixed numbers, the improper fractions will be: \(4 \frac{3}{5} = \frac{(4 \times 5) + 3}{5} = \frac{23}{5}\) \(2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3}\) \(6 \frac{8}{9} = \frac{(6 \times 9) + 8}{9} = \frac{62}{9}\) \(7 \frac{7}{15} = \frac{(7 \times 15) + 7}{15} = \frac{112}{15}\) \(3 \frac{5}{9} = \frac{(3 \times 9) + 5}{9} = \frac{32}{9}\)

Add the improper fractions together

To add fractions, we first need to find a common denominator and then add the numerators. Common denominator = lcm(5, 3, 9, 15, 9) = 45 Now we add the numerators: Sum = \(\frac{23}{5} + \frac{8}{3} + \frac{62}{9} + \frac{112}{15} + \frac{32}{9}\) The equivalent fractions with a common denominator (45) are: Sum = \(\frac{207}{45} + \frac{120}{45} + \frac{310}{45} + \frac{336}{45} + \frac{160}{45}\) Sum = \(\frac{207+120+310+336+160}{45} = \frac{1133}{45}\)

Divide the sum by the total number of mixed numbers

We have 5 mixed numbers, so we divide the sum by 5 to find the average: Average = \(\frac{1133}{45 \times 5} = \frac{1133}{225}\)

Simplify the result

We can convert the improper fraction \(\frac{1133}{225}\) back into a mixed number and choose the correct choice: \(1133 \div 225 = 5\) (remainder = 8) So, the resulting mixed fraction is: \(5 + \frac{8}{225} = 5 \frac{8}{225}\) Therefore, the correct answer is (b) \(5 \frac{8}{225}\).

Key concepts

Mixed Numbers to Improper Fractions

Understanding how to convert mixed numbers to improper fractions is essential when dealing with average calculations that involve mixed numbers. A mixed number consists of a whole number and a fraction combined, such as \(4 \frac{3}{5}\). To work with these, we often need to convert them into improper fractions where the numerator is larger than the denominator.

The conversion formula is straightforward: multiply the whole number by the denominator, add the numerator, then place this value over the original denominator. For instance:\[\frac{(whole \; number \times denominator) + numerator}{denominator}\] Following this approach, \(4 \frac{3}{5}\) becomes an improper fraction:\[\frac{(4 \times 5) + 3}{5} = \frac{23}{5}\] This makes adding and comparing mixed numbers much easier, as they're transformed into a consistent fraction format which can be effectively used in calculations.

Adding Fractions

When it comes to adding fractions, having a common denominator is key. Different denominators are like speaking different languages - to communicate, we need common ground. For fractions, this means we adjust each fraction so that the denominator is the same for all. Once we have a common denominator, we simply add up the numerators to get the sum.

For instance, if you need to add \(\frac{1}{4}\) and \(\frac{2}{5}\), you would first find a common denominator (which in this case would be 20), then convert each fraction:\[\frac{1 \times 5}{4 \times 5} + \frac{2 \times 4}{5 \times 4} = \frac{5}{20} + \frac{8}{20}\] Now, with a shared denominator, you can straightforwardly add the numerators:\[\frac{5 + 8}{20} = \frac{13}{20}\] This way, we can work with fractions that initially seem mismatched and turn them into a single, easy to understand number.

Least Common Multiple (LCM)

Finding the least common multiple (LCM) can sound daunting, but it's just about finding the smallest number that all our denominators can equally divide into. It’s the foundation for adding, subtracting, or comparing fractions with different denominators. Think of it as organizing a party and trying to find a time that works for all guests - LCM is that golden time slot.

To find the LCM of a set of numbers, we can use methods like prime factorization, listing multiples, or the ladder method. For example, if we took the denominators 3, 4, and 5, the LCM would be the smallest number that all three denominators can divide into without a remainder. In this case:\[LCM(3, 4, 5) = 60\]

• 3 divides into 60 (\(60 \div 3 = 20\)),
• 4 divides into 60 (\(60 \div 4 = 15\)),
• 5 divides into 60 (\(60 \div 5 = 12\)).

So, 60 is our LCM, allowing us to work with fractions with denominators of 3, 4, and 5, in a coherent way, without changing the value of the fractions.

Improper Fraction to Mixed Number

Once we've completed calculations with improper fractions, it's often necessary to present the result in a more interpretable form - like a mixed number. Converting an improper fraction to a mixed number can be likened to exchanging a large bill for a combination of smaller bills and coins - it's the same amount, but in a more usable form.

To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient gives us the whole number part, and the remainder becomes the numerator of the fractional part. For example:\[\frac{1133}{225} = 5 \thinspace rem \thinspace 8\] The quotient is 5 (which means \(225 \times 5 = 1125\)), and we have a remainder of 8. We write this remainder over the original denominator to form the fractional part, resulting in a mixed number:\[5 \frac{8}{225}\] With this, we have a mixed number that showcases both the integer and the fractional parts clearly.