A kite company produces kites whose lengths vary directly with their widths. If a kite with length 63 inches has width 48 inches, what is the length of a 40 -inch-wide kite?

Answer 1

If 48 inch wide kite has a length 63 inch,

then a 40 inch wide kite has a length change proportional to width's change, which is,

63 * (40 / 48) = 63 * (5 / 6) = 52.5 inch

Answer 2

The keywords from the setup are "vary directly," so we know we are working with a proportion. Here, we have 63 inches in length correlating to a width of 48 inches, so we can think of this as being 63/48. To create an equivalence, we need "=," and we just have to fill in the remaining information. Think about the proportional relationship:

the length/width of kite 1 must be directly proportional to the length/width of kite 2, or

63/48 = L (for "length")/40

To solve for the unknown, you have one of two options. The first is the "bowtie" method, in which you cross-multiply: 63 * 40 = L * 48, or 2,520 = 48L. Solving algebraically by dividing both sides by 48, we get L = 52.5 inches.

If you were curious, the second method is to find a relationship between the two similar components that are given, in this case the width, to determine a scale factor, understanding that the scale factor must remain consistent between the length and width of one kite to another. To get from a width of 48 inches in kite 1 to a width of 40 inches in kite 2, we can appreciate that kite 2 is 5/6 as large: 48 and 40 are both divisible evenly by 8, the greatest common factor, and dividing both numbers by 8 will yield 6 and 5, meaning the larger kite has 6 "parts" to every 5 "parts" of the smaller kite. In other words, the smaller kite is 5/6 as great in its dimensions as the larger kite, and 48(5/6) = 40. Repeating the process for the other dimension, we get 63(5/6) = 52.5 inches, the same answer as before.

There are other ways to solve the problem, but I will encourage you to explore such possibilities on your own. Flexible thinking in mathematics is often discouraged until college (unless you get someone like me as an ACT/SAT tutor). However, such thinking often leads to more efficient handling of problems and, at the uppermost levels, breakthroughs in mathematics and physics, to mention just two disciplines.

I hope that helps. Good luck with your studies.

Answer 3

If two values are directly proportional, that means they satisfy the equation y = kx for some constant k. So, if we have two existing values of x and y, we can solve for this value of x.

Let's let x be width and y be length. In this example, we have x = 48 and y = 63, so we solve for

64 = k * 48

k = 1.3125

So, if in the next example we have x = 40, then

y = 1.3125 * 40 = 52.5

So that's your length!