Answered

Here are 25 fractions in a table.




ABCDE

1		$\frac{2}{3}$$\frac{2}{5}$$\frac{2}{10}$$\frac{2}{12}$$\frac{2}{100}$

2		$\frac{4}{3}$$\frac{4}{5}$$\frac{4}{10}$$\frac{4}{12}$$\frac{4}{100}$

3		$\frac{7}{3}$$\frac{7}{5}$$\frac{7}{10}$$\frac{7}{12}$$\frac{7}{100}$

4		$\frac{11}{3}$$\frac{11}{5}$$\frac{11}{10}$$\frac{11}{12}$$\frac{11}{100}$

5		$\frac{26}{3}$$\frac{26}{5}$$\frac{26}{10}$$\frac{26}{12}$$\frac{26}{100}$






For each question, be prepared to explain your reasoning.

  1. Identify the greatest fraction in each column (A, B, C, D, and E). ____
  2. Identify the greatest fraction in each row (1, 2, 3, 4, and 5). ____
  3. Which fraction is the greatest fraction in the entire table? ____

Verified Answer :

  • The greatest fraction in each column corresponds to the largest numerator (\(26\)) and are \(\frac{26}{3}, \frac{26}{5}, \frac{26}{10}, \frac{26}{12}, \frac{26}{100}\).
  • The greatest fraction in each row corresponds to the smallest denominator (\(3\)) and are \(\frac{2}{3}, \frac{4}{3}, \frac{7}{3}, \frac{11}{3}, \frac{26}{3}\).
  • The greatest fraction in the entire table is \(\boxed{\frac{26}{3}}\).

Explanation

  1. Data and Problem Analysis

To determine the greatest fraction in each column, row, and the entire table, we first observe the structure of the provided grid of fractions. Each fraction is defined by a numerator \(N\) from the set \(\{2, 4, 7, 11, 26\}\) (representing rows 1 through 5) and a denominator \(D\) from the set \(\{3, 5, 10, 12, 100\}\) (representing columns A through E). The value of any fraction in the table is given by the expression \(\frac{N}{D}\). Comparing fractions with the same denominator allows us to identify the greatest numerator, while comparing fractions with the same numerator allows us to identify the smallest denominator as the indicator of the largest value.

  1. Greatest Fraction in Each Column

In each column, the denominator is held constant. Therefore, the greatest fraction in a column will have the largest numerator.

  • Column A (denominator \(3\)): The fractions are \(\{\frac{2}{3}, \frac{4}{3}, \frac{7}{3}, \frac{11}{3}, \frac{26}{3}\}\). The greatest is \(\frac{26}{3}\).
  • Column B (denominator \(5\)): The fractions are \(\{\frac{2}{5}, \frac{4}{5}, \frac{7}{5}, \frac{11}{5}, \frac{26}{5}\}\). The greatest is \(\frac{26}{5}\).
  • Column C (denominator \(10\)): The fractions are \(\{\frac{2}{10}, \frac{4}{10}, \frac{7}{10}, \frac{11}{10}, \frac{26}{10}\}\). The greatest is \(\frac{26}{10}\).
  • Column D (denominator \(12\)): The fractions are \(\{\frac{2}{12}, \frac{4}{12}, \frac{7}{12}, \frac{11}{12}, \frac{26}{12}\}\). The greatest is \(\frac{26}{12}\).
  • Column E (denominator \(100\)): The fractions are \(\{\frac{2}{100}, \frac{4}{100}, \frac{7}{100}, \frac{11}{100}, \frac{26}{100}\}\). The greatest is \(\frac{26}{100}\).
  1. Greatest Fraction in Each Row

In each row, the numerator is held constant. Therefore, the greatest fraction in a row will have the smallest denominator.

  • Row 1 (numerator \(2\)): The fractions are \(\{\frac{2}{3}, \frac{2}{5}, \frac{2}{10}, \frac{2}{12}, \frac{2}{100}\}\). The greatest is \(\frac{2}{3}\).
  • Row 2 (numerator \(4\)): The fractions are \(\{\frac{4}{3}, \frac{4}{5}, \frac{4}{10}, \frac{4}{12}, \frac{4}{100}\}\). The greatest is \(\frac{4}{3}\).
  • Row 3 (numerator \(7\)): The fractions are \(\{\frac{7}{3}, \frac{7}{5}, \frac{7}{10}, \frac{7}{12}, \frac{7}{100}\}\). The greatest is \(\frac{7}{3}\).
  • Row 4 (numerator \(11\)): The fractions are \(\{\frac{11}{3}, \frac{11}{5}, \frac{11}{10}, \frac{11}{12}, \frac{11}{100}\}\). The greatest is \(\frac{11}{3}\).
  • Row 5 (numerator \(26\)): The fractions are \(\{\frac{26}{3}, \frac{26}{5}, \frac{26}{10}, \frac{26}{12}, \frac{26}{100}\}\). The greatest is \(\frac{26}{3}\).
  1. Greatest Fraction in the Entire Table

To identify the overall maximum value in the table, we compare the greatest fractions found in the columns or the rows. Based on the values determined previously, the column-wise maximums are \(\{\frac{26}{3}, \frac{26}{5}, \frac{26}{10}, \frac{26}{12}, \frac{26}{100}\}\). Since the numerator \(26\) is the largest available and the denominator \(3\) is the smallest available, the fraction \(\frac{26}{3}\) yields the greatest value. This fraction appears in the first column and fifth row. Thus, the greatest fraction in the entire table is \(\frac{26}{3}\).

Examples

This type of grid analysis is fundamental in various real-life scenarios, such as comparing efficiency ratios or pricing across different markets. For instance, if you are analyzing different investment plans where numerators represent potential returns and denominators represent risk levels, identifying the maximum ratio helps in optimizing your selection. By organizing such data into a grid, you can quickly spot the best-performing options relative to your criteria, ensuring you make informed and optimal decisions in fields like finance, engineering, or resource management.

Related Questions