Question

When Angela was cleaning her refrigerator, she found 2 bottles of catsup. Looking at the labels, she noticed that the capacity of the larger bottle was twice the capacity of the smaller bottle. She estimated that the smaller bottle was about \(\frac{1}{3}\) full of catsup and the larger bottle was about \(\frac{2}{3}\) full of catsup. She poured all the catsup from the smaller bottle into the larger bottle. Then, about how full was the larger bottle? A. \(\frac{2}{9}\) full B. \(\frac{1}{2}\) full C. \(\frac{5}{6}\) full D. Completely full E. Overflowing

Short answer

Answer: The larger bottle is 5/6 full.

Step-by-step solution

Evaluate the quantities of catsup in both bottles initially

Let the capacity of the smaller bottle be \(x\). Then, the capacity of the larger bottle is \(2x\). The smaller bottle is \(\frac{1}{3}\) full, which means it contains \(\frac{1}{3}x\) catsup. The larger bottle is \(2*\frac{2}{3}x\) or \(\frac{4}{3}x\) catsup.

Combine the catsup from the smaller bottle into the larger bottle

When Angela pours the smaller bottle into the larger one, we need to sum the two quantities of catsup to find the new quantity of catsup in the larger bottle. The total quantity of catsup in the larger bottle after pouring is \(\frac{1}{3}x+\frac{4}{3}x=\frac{5}{3}x\).

Determine the resulting level of catsup in the larger bottle

To find out how full the larger bottle is, we need to divide the new quantity of catsup in the larger bottle, \(\frac{5}{3}x\), by its total capacity, \(2x\). So, we have \(\frac{\frac{5}{3}x}{2x}\). Simplify the expression by cancelling \(x\): $$ \frac{\frac{5}{3}}{2} = \frac{5}{6}. $$ The larger bottle is now \(\boxed{\frac{5}{6}}\) full, and the correct choice is (C).

Key concepts

Ratio and Proportion

Understanding ratio and proportion is vital when solving problems that involve the relative sizes of two quantities. A ratio is a way to compare two quantities by using division as in miles per hour, where we compare miles and hours. A proportion, on the other hand, is an equation that states two ratios are equivalent.

For example, in the exercise where Angela has two bottles of catsup, we see the use of ratios in the capacity of one bottle being twice that of the other. This establishes a simple ratio of 1:2—the smaller bottle's capacity to the larger bottle's capacity. When she estimates the content levels of the bottles, she's effectively using proportions to relate the part of each bottle filled with catsup to the whole.

By defining the smaller bottle's capacity as 'x', we have set a reference point that allows us to understand and manipulate the quantities involved in proportional terms. The step by step solution makes effective use of these concepts to arrive at the required answer.

Volume and Capacity

Volume and capacity are physical concepts used to describe how much space an object or substance occupies and can be measured in units such as liters, cubic meters, or gallons. In our contextual problem, the volume of catsup each bottle can hold is defined by its capacity.

Volume sometimes is confused with capacity, but while volume is the measure of space, capacity refers to how much a container can hold. Understanding this distinction is crucial when trying to determine the outcome of transferring content from one container to another. Furthermore, being able to estimate and work with these volumes, as Angela did, requires an understanding of how they can be expressed as fractions of the container's capacity and how these fractions interact when combined—the very essence of the exercise provided.

Fractions and Decimals

Fractions and decimals are numerical representations that express parts of a whole. In the context of the given exercise, fractions are used to represent the amount of catsup in each bottle. Fractions can be intimidating, but they become much easier once you understand that the numerator (top number) tells how many parts we have, and the denominator (bottom number) tells how many of those parts make up a whole.

When Angela pours the contents of the smaller bottle into the larger one, we must add the quantities expressed as fractions. The process involves finding a common denominator to combine them, which we can do by ensuring that both quantities are expressed with respect to the same whole. This fundamental understanding of fractions informs every step in solving the presented problem.

Moreover, fractions can often be converted into decimals for simpler computation, especially in scientific and financial settings. However, in our exercise, since the capacities are compared, we maintain the fractional form to clearly visualize the proportions.