Answer:
One row needs to be added to Pattern B to maintain the ratio.
Explanation:
To determine how many rows are added to Pattern B in order to maintain the ratio of straight lines to arcs at 7:6, we need to analyze the initial ratios of straight lines and arcs in Pattern B.
Let's assume Pattern B initially has x straight lines and y arcs. Since the ratio of straight lines to arcs is 7:6, we can set up the equation:
x:y = 7:6
To simplify the equation, we can multiply both sides by a common factor that makes the ratio integers. In this case, the least common multiple (LCM) of 7 and 6 is 42. Multiplying both sides by 42, we get:
42x:42y = 7:6
Simplifying further, we have:
6x = 7y
Now we need to find a solution for x and y that satisfies this equation. The simplest solution is x = 7 and y = 6, which gives us the initial ratio for Pattern B.
To maintain the same ratio when adding more rows, we need to add an equal number of straight lines and arcs in each row. Since the initial ratio is 7:6, we need to add 7 straight lines and 6 arcs in each additional row.
Therefore, to determine how many rows need to be added, we can divide the difference in the number of straight lines and arcs by the number of straight lines and arcs added per row:
Difference = 7x - 6y
We want the difference to be zero, meaning an equal number of straight lines and arcs. Therefore:
7x - 6y = 0
We can now substitute x = 7 and y = 6 into the equation:
7(7) - 6(6) = 49 - 36 = 13
The difference is 13. Since we add 7 straight lines and 6 arcs per row, the difference of 13 indicates that we need to add 13 / (7 + 6) = 1 row to maintain the ratio of 7:6.
Hence, one row needs to be added to Pattern B to maintain the ratio of straight lines to arcs at 7:6.