Question

Translate to an equation and solve. The nutrition label on a cereal bar shows that one bar has 3.5 grams of total fat and that this is \(6 \%\) of the recommended daily value. Based on the label, how many grams of fat are recommended?

Short answer

Answer: Approximately 58.33 grams.

Step-by-step solution

Understand the given information

We are given that one cereal bar has 3.5 grams of fat, which is 6% of the recommended daily value.

Set up the proportion

To calculate the recommended daily value, we can set up a proportion with the percentage (6%) and the known weight (3.5 grams): \(\frac{3.5}{x} = \frac{6}{100}\) Here, x is the total recommended grams of fat per day.

Solve for x

To solve for x, we can cross-multiply: \(3.5 \cdot 100 = 6 \cdot x\)

Calculate x

Divide both sides by 6 to isolate x: \(x = \frac{3.5 \cdot 100}{6}\)

Find the recommended daily value

Now, we can calculate the value of x: \(x = \frac{3.5 \cdot 100}{6} = \frac{350}{6} = 58.33\) Based on the label, there are approximately 58.33 grams of fat recommended per day.

Key concepts

Understanding Percentage Problems

Percentage problems are a fundamental aspect of mathematics, often found in various practical scenarios like shopping discounts, nutrition facts, and statistical data. The key to solving percentage problems is to recognize that 'percent' means per hundred. So, when we say 6%, we mean 6 out of every 100.

Let's break down the percentage problem in our exercise. We're told that 3.5 grams of fat represents 6% of the recommended daily value. To find the full 100%, a simple proportion can be established. A general equation for percentage problems is: \( \frac{\text{Part}}{\text{Whole}} = \frac{\text{Percentage}}{100} \). In this case, the part is the known quantity (3.5 grams) and the whole is the total amount we're trying to find. Solving such problems becomes a matter of isolating the variable and calculating its value.

Solving Proportion Problems

Proportion problems involve finding a missing number in a set of ratios that are equal to each other. Ratios compare two quantities, and proportions state that two ratios are equal. The example problem provided is classic: \( \frac{3.5}{x} = \frac{6}{100} \), where we're comparing grams of fat to the recommended daily value, which is unknown.

To solve for the unknown value, a common strategy called 'cross-multiplication' is used: the product of the means equals the product of the extremes. In our case, multiply 3.5 by 100 and 6 by x, then solve for x: \( 3.5 \times 100 = 6 \times x \). Dividing both sides by 6 yields the recommended daily value. Understanding how these proportions work is critical for solving not only math problems but also interpreting real-world situations.

Navigating Math Word Problems

Math word problems are perceived by many students as complex puzzles. They integrate real-life scenarios with mathematical concepts, requiring both comprehension and computation skills. The first step toward solving them is to identify the relevant information and discard any superfluous details.

Our exercise deals with nutrition labels and recommends daily values, embedding a proportion problem within a word problem. By translating words into equations—such as converting '6% of the recommended daily value' to the equation \( \frac{3.5}{x} = \frac{6}{100} \)—we transform the narrative into a solvable mathematics question. Practicing these problems enhances problem-solving skills and helps students apply math to everyday life.