Question

Sandi has brewed 55 ounces of iced tea. How many full glasses of tea can be filled if it takes \(7 \frac{3}{4}\) ounces of tea to fill each glass? A. 47 glasses B. 8 glasses C. 7 glasses D. 6 glasses

Short answer

C. 7 glasses

Step-by-step solution

Convert Mixed Number to Improper Fraction

First, convert the mixed number for the ounces of tea per glass into an improper fraction. For the mixed number \(7 \frac{3}{4}\), multiply the whole number 7 by the denominator 4 and add the numerator 3: \(7 \times 4 + 3 = 28 + 3 = 31\). So, \(7 \frac{3}{4} = \frac{31}{4}\).

Set Up the Division

Divide the total amount of iced tea by the amount needed to fill one glass. This means dividing 55 ounces by \(\frac{31}{4}\). When dividing by a fraction, multiply by its reciprocal: \(55 \div \frac{31}{4} = 55 \times \frac{4}{31}\).

Perform the Multiplication

Calculate the multiplication: \(55 \times \frac{4}{31} = \frac{55 \times 4}{31} = \frac{220}{31}\).

Simplify the Fraction

Simplify \(\frac{220}{31}\): Divide 220 by 31 to get approximately 7.09.

Determine the Number of Full Glasses

Since the answer must be a whole number count of full glasses, we take the integer part (the whole number) of the division result. The integer part of 7.09 is 7.

Final Step: Select the Correct Answer

Compare the integer result to the given options. The closest match is option C. Therefore, Sandi can fill 7 full glasses of tea.

Key concepts

Improper Fractions

Understanding improper fractions is crucial when working with mixed numbers. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, follow these steps:

First, multiply the whole number by the denominator. For example, in the mixed number \(7 \frac{3}{4}\), multiply 7 by 4. Then, add the numerator to the product obtained. Here, we add 3 to the resultant product, 28, to get 31. This results in the improper fraction \(\frac{31}{4}\).

Reciprocal Multiplication

When dividing by a fraction, it is helpful to multiply by the reciprocal, which is the inverse of the fraction. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

In the exercise, we're asked to divide 55 by \(\frac{31}{4}\). By multiplying instead by the reciprocal of \(\frac{31}{4}\), which is \(\frac{4}{31}\), we change the division problem into a multiplication problem.

This makes the equation easier to solve: \(55 \div \frac{31}{4} = 55 \times \frac{4}{31}\). This step simplifies the calculation process substantially.

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form. This step helps in understanding and comparing fractions more easily. After performing the multiplication, you get \(\frac{220}{31}\). This fraction cannot be simplified further as 220 and 31 do not have common factors apart from 1.

Therefore, we move to converting this improper fraction into a more usable form for the problem at hand. In this case, we convert it to a decimal to determine how many full glasses can be filled. \(\frac{220}{31} \approx 7.09\).

Integer Division

To determine how many whole glasses can be filled, we focus on the integer part of the division result obtained from the fraction simplification. Integer division discards the decimal part and retains only the whole number.

Since we need the number of full glasses Sandi can fill, we take the integer part of 7.09, which is 7. This means that 7 full glasses can be filled with 55 ounces of iced tea, using \(7 \frac{3}{4}\) ounces per glass.

This concept helps in solving many real-life problems where partial results are not feasible or practical.