Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Jan 10, 2024

Write an equation to describe the proportional relationship between paint used (x quarts) and wall area (y sq. yards) given the table: $3x = 23 \frac{1}{2}y, 4x = 31 \frac{1}{3}y, 5x = 39 \frac{1}{6}y$ .

STEP 1

Assumptions
1. The relationship between the amount of paint used and the area covered is proportional.
2. We have three pairs of values representing this relationship: (3, $23 \frac{1}{2}$ ), (4, $31 \frac{1}{3}$ ), and (5, $39 \frac{1}{6}$ ).
3. We need to find the constant of proportionality $k$ such that $y = kx$ .
4. Once we find $k$ , we can write the equation that describes the relationship between $x$ and $y$ .

STEP 2

First, we need to convert the mixed numbers into improper fractions to make calculations easier.
For $23 \frac{1}{2}$ : $23 \frac{1}{2} = 23 + \frac{1}{2} = \frac{46}{2} + \frac{1}{2} = \frac{46 + 1}{2} = \frac{47}{2}$
For $31 \frac{1}{3}$ : $31 \frac{1}{3} = 31 + \frac{1}{3} = \frac{93}{3} + \frac{1}{3} = \frac{93 + 1}{3} = \frac{94}{3}$
For $39 \frac{1}{6}$ : $39 \frac{1}{6} = 39 + \frac{1}{6} = \frac{234}{6} + \frac{1}{6} = \frac{234 + 1}{6} = \frac{235}{6}$

STEP 3

Now we can calculate the constant of proportionality $k$ using one of the pairs of values. We will use the first pair (3, $23 \frac{1}{2}$ ) or (3, $\frac{47}{2}$ ).
$k = \frac{y}{x} = \frac{\frac{47}{2}}{3}$

STEP 4

Calculate the value of $k$ .
$k = \frac{\frac{47}{2}}{3} = \frac{47}{2 \times 3} = \frac{47}{6}$

STEP 5

Now that we have the constant of proportionality $k$ , we can write the equation that describes the relationship between the amount of paint used $(x)$ and the area of the wall that can be painted $(y)$ .
$y = kx$

STEP 6

Substitute the value of $k$ into the equation.
$y = \frac{47}{6}x$
This is the equation that describes the relationship between the amount of paint used and the area of the wall that can be painted.

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