Problem 24 - Prealgebra

Question

Model Trains The size of a model train relative to an actual train is referred to as its scale. Each scale is associated with a ratio as shown in the table. For example, an HO model train has a ratio of 1 to 87, meaning it is \(\frac{1}{87}\) as large as an actual train. Nutrition If 125 grams of peas contain 26 grams of carbohydrates, how many grams of carbohydrates do 375 grams of peas contain?

Step-by-Step Solution

Answer

375 grams of peas contain 78 grams of carbohydrates.

Step 1: Understand the Problem

We need to find how many grams of carbohydrates are present in 375 grams of peas, given that 125 grams of peas contain 26 grams of carbohydrates. This means we will set up a proportion to solve for the unknown carbohydrate amount.

Step 2: Set Up the Proportion

We are given that 125 grams of peas contain 26 grams of carbohydrates. We need to find out how many grams (\(x\)) are in 375 grams of peas. Set up the proportion: \[ \frac{125}{26} = \frac{375}{x} \]

Step 3: Solve the Proportion

Cross-multiply to solve for \(x\). This gives us: \[ 125 \times x = 375 \times 26 \]Simplify the right-hand side: \[ x = \frac{375 \times 26}{125} \]

Step 4: Calculate the Value of x

Perform the multiplication and division:- Multiply: \(375 \times 26 = 9750\)- Divide: \(\frac{9750}{125} = 78\)Thus, \(x = 78\), which means 375 grams of peas contain 78 grams of carbohydrates.

Key Concepts

Ratios

A ratio is a way to compare two quantities by using division. If you're dealing with a model train that is \(1:87\), it means that every 1 unit of the model train is equivalent to 87 units of the actual train. Ratios are incredibly useful for scaling problems. They help us understand how large or small something is in comparison to something else.
Here's how you might see ratios in real life:

• Cooking: Ingredients often use ratios to maintain taste consistency.
• Maps: To understand distances between two places, map scales often use ratios.

In mathematics, understanding ratios is fundamental for setting up proportions, which enable us to solve problems involving similar relationships. A ratio can also be presented in fractional form, like \(\frac{1}{87}\). This fractional representation is convenient when setting up and solving mathematical equations.

Cross-Multiplication

Cross-multiplication is a technique for solving equations where two ratios are set equal to each other, often called a proportion. This is immensely helpful when you're attempting to find an unknown value. Here's the simple process:
When given a proportion in the form \[ \frac{a}{b} = \frac{c}{d} \], cross-multiplication involves multiplying across the "X" shape that forms: \

Multiply \(a\) by \(d\)

Multiply \(b\) by \(c\)

Equating these products gives: \ a \times d = b \times c \, \[ \Rightarrow \]\, What makes cross-multiplication handy is that it simplifies solving for unknown variables in proportional relationships. Remember to never cross-multiply when your proportion isn't set up correctly; always ensure ratios are comparable, meaning their units or measures are aligned.

Word Problems

Word problems present mathematical concepts in a real-world context, often leading students to apply math in practical ways. Solving word problems involves translating a description or story into a mathematical statement. To tackle word problems effectively:

• Identify what you know and what you need to find.
• Translate the problem into a mathematical equation.
• Solve the equation using appropriate methods.
• Check your solution for accuracy and reasonableness.

Take the given problem involving peas and carbohydrates: we want to find out how many carbohydrates are in 375 grams of peas. We had the information for 125 grams, and used proportion, \[ \frac{125}{26} = \frac{375}{x} \], to find the unknown. Such practical exercises enhance one's problem-solving skills and provide a clear method for using mathematics to answer real-life questions.