Question

Ten floorboards with equal widths laid down side-to-side cover a width of approximately \(7 \frac{3}{4}\) feet. At this rate, which of the following is the closest to the number of boards laid side-to-side needed to cover a width of 32 feet? A) 15 B) 20 C) 30 D) 40

Short answer

The closest number of boards laid side-to-side needed to cover a width of 32 feet is 41 (Answer D).

Step-by-step solution

Find the width of one floorboard

We know that 10 floorboards cover a width of \(7 \frac{3}{4}\) feet. To find the width of one floorboard, we can divided the total width by the number of floorboards: Width of one floorboard = \(\frac{Width\:covered\:by\:10\:floorboards}{Number\:of\:floorboards}\) = \(\frac{7 \frac{3}{4}}{10}\) Let's first convert \(7 \frac{3}{4}\) into an improper fraction: \(7 \frac{3}{4}\) = \(\frac{7 \times 4 + 3}{4}\) = \(\frac{31}{4}\) Now plug it into our formula: Width of one floorboard = \(\frac{\frac{31}{4}}{10}\) = \(\frac{31}{40}\) feet

Find the number of floorboards needed to cover 32 feet

Let x be the number of floorboards needed to cover 32 feet width: x floorboards * Width of one floorboard = 32 feet x * \(\frac{31}{40}\) = 32 Now, we need to solve for x: x = \(\frac{32}{\frac{31}{40}}\) = \(32 * \frac{40}{31}\) To find the closest integer value to the number of floorboards needed (since we cannot have a fraction of a floorboard), round this result to the nearest whole number: x ≈ \(32 * \frac{40}{31}\) = 41.29 ≈ 41 The closest number of boards laid side-to-side needed to cover a width of 32 feet is 41 (Answer D).

Key concepts

Ratio and Proportion

The concepts of ratio and proportion are pivotal in various mathematical problems, including those on the SAT. A ratio is a way to compare two quantities by dividing them to show how many times one value contains the other. For example, if there are 2 apples for every 3 oranges, the ratio of apples to oranges is 2:3.

Proportion, on the other hand, is an equation that states that two ratios are equivalent. When we say that one floorboard covers a certain fraction of the total width needed, we are setting up a relationship between the number of floorboards and the total width, which in mathematical terms, is a proportion.

To solve the given problem, understanding that each floorboard's width is in a fixed proportion to the total width covered by multiple floorboards was essential. By determining the width of a single board, we could calculate the number of boards needed for a new total width. This exercise applies ratio and proportion to deduce the number of floorboards required to cover a given width, exemplifying how these concepts are used in real-world and SAT problems.

SAT Practice Questions

SAT math problems are designed to evaluate a student's reasoning and problem-solving abilities in a timed environment. The questions range from algebra, geometry, to more advanced topics like ratio and proportion. The exercise presented here is a typical example of what students might encounter on the SAT.

It’s crucial for students to understand how the SAT formats these questions and to practice with examples that reflect the actual test. In our floorboard question, the SAT is assessing the student's ability to apply mathematical concepts to solve a real-world scenario involving proportional reasoning.

Exercise Improvement Advice

When engaging with SAT practice questions, breaking down the problem into manageable steps can greatly enhance understanding. For the floorboards example, isolating the step of converting the mixed number into an improper fraction helps clarify the process. Students should also practice rounding numbers, as in the final step of this problem, where the answer must be a whole number, indicative of a real life scenario where partial floorboards are not an option.

Arithmetical Reasoning

Arithmetical reasoning involves the ability to apply basic arithmetic operations like addition, subtraction, multiplication, and division to solve problems. This reasoning extends to understanding and working with fractions, decimals, and percentages. In our textbook exercise, this type of reasoning was critical in every step.

From converting fractions to solving the equation for the unknown quantity, the problem required arithmetical manipulation and clear understanding of the rules governing arithmetic operations. Furthermore, arithmetical reasoning includes estimation skills showcased when finding the nearest whole number of floorboards needed to cover a given width, which also involves understanding the practical constraint that floorboards come in whole units.

Arithmetical reasoning is not just about performing computational tasks but also involves logical thinking to interpret and solve problems. Exercises such as this floorboard problem provide students with opportunities to refine their arithmetic skills while also learning to apply them in context.