Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(2 \frac{7}{8} \times 3 \frac{1}{4}=\) ______

Short answer

The product is \(9 \frac{11}{32}\).

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. For the mixed number \(2 \frac{7}{8}\):- Multiply the whole number 2 by the denominator 8: \(2 \times 8 = 16\)- Add the numerator 7: \(16 + 7 = 23\)- So, \(2 \frac{7}{8} = \frac{23}{8}\)For the mixed number \(3 \frac{1}{4}\):- Multiply the whole number 3 by the denominator 4: \(3 \times 4 = 12\)- Add the numerator 1: \(12 + 1 = 13\)- So, \(3 \frac{1}{4} = \frac{13}{4}\)

Multiply the Improper Fractions

Now we can multiply the two improper fractions \(\frac{23}{8}\) and \(\frac{13}{4}\).Use the formula: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)Substitute the values:\(\frac{23}{8} \times \frac{13}{4} = \frac{23 \times 13}{8 \times 4}\)Perform the multiplications:\(23 \times 13 = 299\)\(8 \times 4 = 32\)So, \(\frac{23}{8} \times \frac{13}{4} = \frac{299}{32}\).

Simplify the Fraction

Check if the fraction \(\frac{299}{32}\) can be reduced to its lowest terms.Since 299 and 32 have no common factors (other than 1), \(\frac{299}{32}\) is already in its simplest form.

Convert the Improper Fraction to a Mixed Number

Divide 299 by 32 to convert it back to a mixed number.- 299 divided by 32 gives a quotient of 9 and a remainder of 11.- So, the mixed number is \(9 \frac{11}{32}\).Thus, \(\frac{299}{32} = 9 \frac{11}{32}\).

Key concepts

mixed numbers

Mixed numbers are fractions that consist of a whole number and a proper fraction combined, such as \(2 \frac{7}{8}\). They are useful for representing quantities greater than one in a more relatable way. When dealing with fraction multiplication, it's important to first convert mixed numbers into improper fractions.

To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the fraction's denominator.
• Add the numerator to this product.
• The result becomes the numerator of the improper fraction, with the denominator remaining the same.

For example, when converting \(2 \frac{7}{8}\), you'd multiply \(2 \times 8 = 16\), add \(7\) to get \(23\), resulting in \(\frac{23}{8}\). Practicing this conversion helps when multiplying mixed numbers, as it simplifies the process.

improper fractions

Improper fractions have numerators larger than their denominators, like \(\frac{23}{8}\). These fractions are useful in fraction multiplication because they simplify arithmetic operations. An improper fraction signifies a value more than one whole, making calculations straightforward.

When multiplying fractions, simply multiply the numerators together and the denominators together. Using improper fractions avoids the tedious work of having to separately account for whole numbers, as seen:

• \(\frac{23}{8} \times \frac{13}{4} = \frac{23 \times 13}{8 \times 4} = \frac{299}{32}\)

Improper fractions are especially handy in ensuring each step is clear and precise, leading you directly to the answer.

simplifying fractions

Simplifying fractions is the process of reducing a fraction to its lowest terms, ensuring the smallest possible numerator and denominator. This step is crucial in presenting the final answer in an understandable form.

To simplify fractions, it’s important to determine if the numerator and denominator have any common factors beyond 1. If they do, divide both by the greatest common factor (GCF) to simplify. In our exercise, \(\frac{299}{32}\) was already in its simplest form because 299 and 32 share no common factors. When simplifying, aim for clarity in your final result.

converting fractions

Converting between mixed numbers and improper fractions is commonly required in mathematical computations, especially when handling multiplication or division. It is essential to ensure fractions are in the form most suitable for the calculation at hand.

After computing the multiplication of two improper fractions, the final step often involves converting the improper fraction back into a mixed number for ease of understanding. When converting \(\frac{299}{32}\) into a mixed number, divide 299 by 32:

• The quotient (9) is the whole number part of the mixed number.
• The remainder (11) becomes the numerator of the fraction part.
• The denominator remains the same (32).

Thus, \(\frac{299}{32} = 9 \frac{11}{32}\). Being adept at these conversions enriches your problem-solving skills, allowing you to present your answers in diverse forms.