Question

Mixture A farmer mixes gasoline and oil to make 2 gallons of mixture for his two-cycle chain saw engine. This mixture is 32 parts gasoline and 1 part two- cycle oil. How much gasoline must be added to bring the mixture to 40 parts gasoline and 1 part oil?

Short answer

The farmer needs to add approximately \(\frac{16}{33}\) or 0.48 gallons of gasoline to achieve a 40:1 gasoline to oil ratio.

Step-by-step solution

Establish the Initial and Desired Ratios

Initially, the mixture was 32 parts gasoline to 1 part oil, giving a ratio of 32:1. The desired ratio is 40 parts gasoline to 1 part oil, or 40:1.

Find the Total Parts in the Initial Mixture

The total parts in the initial mixture is 32 parts gasoline + 1 part oil = 33 parts.

Calculate the Gasoline part in Gallons

Since the total 2 gallons of mixture represented 33 parts, each part is equivalent to \(\frac{2}{33}\) gallons. Thus, the gasoline part of the mixture, which is 32 parts, is equivalent to \(32 \times \frac{2}{33} = \frac{64}{33}\) gallons.

Compute the Required Gasoline for the Desired Ratio

In order to achieve the ratio of 40:1 whilst keeping the oil constant, we calculate the required gasoline in gallons. The oil part is still 1 and corresponds to \( \frac{2}{33} \) gallons. Hence, the gasoline required for a 40:1 ratio is \(40 \times \frac{2}{33} = \frac{80}{33} \) gallons.

Obtain the Additional Gasoline needed

Subtract the initial gasoline quantity from the required gasoline quantity to get the additional gasoline needed: \( \frac{80}{33} - \frac{64}{33} = \frac{16}{33} \) gallons is the additional quantity of gasoline needed to change the mixture to a 40:1 ratio.

Key concepts

Ratio and Proportion

Understanding the concept of ratio and proportion is essential when working with mixtures in algebraic problems. A ratio is a way to compare two quantities by dividing one by the other to hand us an insight into how they relate to each other. For instance, if a mixture contains 32 parts gasoline and 1 part oil, as in our farmer's dilemma, we write this ratio as 32:1, indicating that for every 1 part of oil, there are 32 parts of gasoline.

Proportions, on the other hand, tell us that two ratios are equal. It's like saying, 'If the mixture has a pattern of so much gasoline to oil, how much of each will I need to keep that pattern but increase the total amount?' When our farmer wants to increase the gasoline to be 40 parts while keeping oil at 1 part, he uses a proportion: the new ratio 40:1 should be equal to the old ratio, just with more gasoline. By using this concept, he can work out exactly how much more gasoline is needed without changing the amount of oil in the mixture.

Parts Per Whole Concept

The 'parts per whole' concept is integral for solving mixture problems. This idea is used to understand what fraction of the whole each part of the mixture represents. In our problem, when we say there are 33 parts in total (32 parts gasoline and 1 part oil), each part represents a fraction of the whole mixture, which is 2 gallons in this case.

To find out how much each part is in gallons, we divide the total volume of the mixture by the total number of parts. This operation is expressed as \(\frac{2\text{ gallons}}{33\text{ parts}}\), giving us the volume of each part. Understanding each part's volume is necessary to maintain the correct ratio when adjusting the mixture. By applying this parts per whole calculation, the farmer can determine how to modify his 32:1 mixture to obtain the desired 40:1 ratio.

Algebraic Expressions

Algebraic expressions are equations that contain numbers, variables, and arithmetic operations. They are a vital tool in an algebraist's toolkit and are essential in mixture problems to represent the relationships between different parts of the mixture.

In our farmer's situation, we use algebraic expressions to represent the initial volume of gasoline and oil, the desired volumes, and the additional gasoline needed. The expressions involve multiplication and subtraction to calculate the exact amount of gasoline. For example, to find the additional gasoline needed when changing the ratio, an algebraic expression is used: \(\frac{80}{33} - \frac{64}{33}\). This expression helps us to derive that \(\frac{16}{33}\) gallons of gasoline is required to reach the 40:1 mixture ratio. By manipulating algebraic expressions, students can solve for unknown quantities and gain a deeper understanding of how to manage proportions in mixtures.