Problem 45 - Introductory Chemistry Atoms First

Question

It takes six cups of flour to bake one cake; exactly one cup of flour has a mass of \(120.0 \mathrm{~g}\). If you have \(6955 \mathrm{~g}\) of flour, how many cakes can you bake? (Hint: Two conversion factors are given in this problem. Find them and write them down first in ratio form. Then use them in the correct form with the measured quantity, which is \(6955 \mathrm{~g}\) of flour.)

Step-by-Step Solution

Answer

With 6955 g of flour, we can bake 9 cakes.

Step 1: Identify the conversion factors

The first conversion factor is the relationship between grams and cups: \(1 \: \text{cup} = 120.0 \: \text{g}\) The second conversion factor is the relationship between cups and cakes: \(1 \: \text{cake} = 6 \: \text{cups}\) Now we will use these conversion factors to convert the given mass of flour to the number of cakes.

Step 2: Convert grams of flour to cups

Using the grams-to-cups conversion factor, we have: \(\frac{1 \: \text{cup}}{120.0 \: \text{g}}\) We will multiply this factor by the given mass of flour to convert it to cups: \(6955 \: \text{g} \times \frac{1 \: \text{cup}}{120.0 \: \text{g}}\) Dividing 6955 by 120 and calculating: \(\text{Number of cups} = \frac{6955}{120} = 57.9583 \: \text{cups}\)

Step 3: Convert cups of flour to cakes

Next, we will use the cups-to-cakes conversion factor: \(\frac{1 \: \text{cake}}{6 \: \text{cups}}\) Multiply this factor by the number of cups obtained in the previous step: \(57.9583 \: \text{cups} \times \frac{1 \: \text{cake}}{6 \: \text{cups}}\) Dividing 57.9583 by 6 and calculating: \(\text{Number of cakes} = \frac{57.9583}{6} = 9.6597 \: \text{cakes}\) Since we cannot bake a fraction of a cake, we'll round this down to the nearest whole number:

Step 4: Round down to the nearest whole number of cakes

The total number of cakes that can be baked with 6955 g of flour is: \(\text{Number of cakes} = \lfloor 9.6597 \rfloor = 9 \: \text{cakes}\) So, with 6955 g of flour, we can bake 9 cakes.

Key Concepts

Conversion Factors Unveiled

Conversion factors are like magic bridges that help us move from one unit of measurement to another. In this homework problem, we have two important conversion factors. The first one is the relationship between grams and cups of flour:\(1 \: \text{cup} = 120.0 \: \text{g}\). This tells us how many grams make up one cup. The second conversion factor connects cups to cakes, shown as:\(1 \: \text{cake} = 6 \: \text{cups}\). This means we need six cups to bake a single cake. To solve the problem, you start by getting the base ingredients in grams, change them to cups, and then from cups to cakes using these bridges.

Understanding Measurement Units

Measurement units like grams, cups, and cakes communicate the amount of something you have or need. Let's break them down.

• Grams (g): A gram is a metric unit of mass, and in this recipe problem, it helps measure the weight of flour.
• Cups: A cup is a common unit used in recipes to measure volume. It helps understand the quantity in a way that's tangible when cooking.
• Cakes: In this problem, a cake represents the final product you achieve by using your ingredients.

Measurement units go hand in hand with conversion factors to guide how you transform and use different amounts effectively in tasks and recipes.

Exploring Mass and Weight

When we talk about mass and weight in everyday situations, like baking, they seem interchangeable, but they are not.

• Mass: It represents the amount of matter in an object and is measured in grams in this context. Mass doesn't change whether you’re on Earth or the Moon.
• Weight: It's actually the force exerted by gravity on an object. In this problem, the focus is more on mass because it's consistent and helps determine the flour amount precisely.

While weight might vary due to gravity, mass stays constant, and in the world of baking, it's this steady mass that tells us how much flour we truly have.

Rounding Numbers with a Purpose

Rounding numbers plays a crucial role, especially in practical problems like baking. When applying conversion factors, you often end up with decimal values.
For example, after converting flour in grams to cups, and then to cakes, you might get a result like \(9.6597\) cakes. But practically, you can’t bake part of a cake. Here’s where rounding comes in.

• Round down: When dealing with whole items (like cakes), it's best to round down to ensure you don't overestimate what you can make.
• Rounding rules: Generally, anything less than 0.5 means round down to the nearest whole number, providing an accurate and feasible solution.

In the solution above, rounding down from \(9.6597\) ensures you know you can safely bake 9 whole cakes without overpromising to your cakeworthy audience.