Question

Priya is planning to send her favorite dry rub recipe to a friend who lives in France. Before sending the recipe, Priya wants to convert the American customary units in the instructions into metric units so that her friend will easily be able to understand the measurements. If the recipe calls for a ratio of four ounces of paprika to every seven ounces of chili powder, and if Priya's friend is planning to make a large batch of dry rub with 91 total ounces of chili powder, approximately how many total grams of paprika and chili powder will the recipe require? ( 1 ounce \(=28.3\) grams) A) 4,047 grams B) 4,521 grams C) 4,925 grams D) 5,149 grams

Short answer

A) 4,047 grams

Step-by-step solution

Convert the ratio to grams

First, we need to convert the given ratio in ounces into a ratio in grams. We are given that 1 ounce is equal to 28.3 grams. By multiplying the amount of paprika and chili powder in ounces by their conversion factor, we can obtain the ratio in grams: Paprika: \(4 \,\text{ounces} \cdot 28.3 \,\frac{\text{grams}}{\text{ounce}} = 113.2\,\text{grams}\) Chili powder: \(7\,\text{ounces} \cdot 28.3 \,\frac{\text{grams}}{\text{ounce}}=198.1\,\text{grams}\) So the ratio of paprika to chili powder in grams is approximately \(113.2:198.1\).

Find the amount of paprika for the large batch

We know that there are 91 ounces of chili powder in the large batch. We need to find out how many grams of paprika are needed for that amount of chili powder. We can set up a proportion using the ratio \(113.2:198.1\): \(\frac{113.2\,\text{grams}}{198.1\,\text{grams}}=\frac{\text{grams}\,\text{of}\,\text{paprika}}{91\,\text{ounces}}\) We can cross-multiply to solve for the grams of paprika: \((113.2\,\text{grams})\cdot(91\,\text{ounces})=(198.1\,\text{grams})\cdot(\text{grams}\,\text{of}\,\text{paprika})\) \(10,291.2\,\text{gram}\cdot\text{ounces}=198.1\,\text{grams}\cdot(\text{grams}\,\text{of}\,\text{paprika})\) Now, divide by 198.1 grams to find the grams of paprika: \(\frac{10,291.2\,\text{gram}\cdot\text{ounces}}{198.1\,\text{grams}}=(\text{grams}\,\text{of}\,\text{paprika})\) \(\Rightarrow 51.9\,\text{ounces}=\text{grams}\,\text{of}\,\text{paprika}\)

Convert the total amount to grams

Now we have the amount of paprika required for the large batch in ounces (51.9 ounces) and the amount of chili powder (91 ounces). We need to find the total amount in grams: Paprika: \(51.9\,\text{ounces} \cdot 28.3 \,\frac{\text{grams}}{\text{ounce}} = 1468.57\,\text{grams}\) Chili powder:_91\,\text{ounces} \cdot 28.3 \,\frac{\text{grams}}{\text{ounce}} = 2573.3\,\text{grams}\)_Now,_add_up_the_two_amounts_factors_to_get_the_total_amount_of_both_paprika_and_chili_powder_required_in_grams:_2get_the_total_amount_of_both_paprika-+ Total grams: \(1468.57\,\text{grams} + 2573.3\,\text{grams} = 4041.87\,\text{grams}\)

Select the approximate total

The calculated total amount needed is 4041.87 grams. Look at the given options: A) 4,047 grams B) 4,521 grams C) 4,925 grams D) 5,149 grams The approximate total of 4041.87 grams is closest to 4,047 grams. So the correct answer is: A) 4,047 grams

Key concepts

Unit Conversion

Understanding unit conversion is vital in many areas of study and everyday life. It refers to changing the measurement of a quantity from one unit to another without altering its value. For instance, when Priya wants to share her recipe with a friend in France, she needs to convert the U.S. customary units to metric units because different countries use different measurement systems.

The process of converting units is usually done by multiplying the original measurement by a conversion factor, which represents the relationship between two different units. In the context of the recipe, the conversion factor is 28.3 grams per ounce. This means one ounce is equivalent to 28.3 grams. This factor allows us to convert ounces to grams by mere multiplication, a simple yet powerful aspect of unit conversion.

Relevance in Real-Life Situations

Understanding unit conversion is not only academically important but practical as well. Knowing how to convert measurements properly is crucial when traveling, cooking, purchasing materials, and even in understanding weather reports. Improved comprehension also prevents confusion and errors that could result from misinterpreting quantities.

Measurement in Math

Measurement in math is the process of determining the size, length, or amount of something, typically expressed in standard units. It's a fundamental concept that serves as the foundation for numerous applications in various scientific and engineering disciplines, as well as in everyday life.

For students, mastering measurement is not just about memorizing formulas and conversion factors; it's also about understanding the underlying concepts, such as the relationship between units, the dimensionality of measures, and the precision and accuracy required in different contexts.

Importance of Precision in Measurements

Precision when measuring ensures accuracy, which is paramount in fields such as medicine, where exact dosages can be crucial, or engineering, where the integrity of structures can rely on precise measurements. The challenge in the exercise provided isn't just in the mathematical conversion but in understanding the real-world implications of measurement and unit conversion, which are essential skills students can apply outside academic environments.

SAT Math Preparation

The SAT Math section is designed to assess a student’s proficiency and understanding of mathematical concepts ranging from arithmetic to algebra and advanced problem-solving. A key area within SAT math preparation is mastering measurement and unit conversion. This skill is required not only to solve particular math problems but also to understand the context of questions and apply the correct mathematical principles.

Thorough preparation for these topics involves familiarizing oneself with different units of measurement and their conversion factors, practicing problems similar to the one encountered by Priya and her recipe, and developing strategies to efficiently convert and solve measurement problems within the tight time constraints of the exam.

Strategies for Effective SAT Math Prep

To excel in the SAT Math section, students should incorporate practice problems that involve conversion and real-world applicability, just like the recipe scenario. Using practice tests, reviewing concepts, and managing time effectively are all strategies that contribute to improving performance on the SAT math section. By integrating unit conversions into study routines, students enhance their mathematical flexibility, a skill that the SAT—and many real-life situations—often demands.