Question

practice understanding proportional relationships
study how the example shows a proportional relationship.
then solve problems 1 - 5.
example
the double number line shows the pounds of clay jamal uses to make different numbers of identical bowls. is there a proportional relationship between the pounds of clay and the number of bowls? if so, what is the constant of proportionality for pounds of clay per bowl?
all of the ratios for clay : bowls are equivalent. so, there is a proportional relationship.
all of the ratios have the same rate, \\( \frac { 2 } { 3 } \\) pound of clay per bowl. the constant of proportionality is \\( \frac { 2 } { 3 } \\).
1 a. look at the example. write an equation that shows the relationship between the pounds of clay jamal uses, c, and the number of bowls he makes, b.
b. explain how to find the pounds of clay jamal needs to make 8 bowls.
2 zara has paper rectangles of different sizes. every rectangle is 5 cm longer than it is wide. is there a proportional relationship between the lengths and widths of these rectangles? explain.

Explanation:

1a

Step1: Recall proportional relationship formula

For a proportional relationship, the formula is \( c = k \times b \), where \( k \) is the constant of proportionality. From the example, the constant of proportionality (pounds of clay per bowl) is \( \frac{2}{3} \).

Step2: Substitute \( k \) into the formula

Substitute \( k = \frac{2}{3} \) into \( c = k \times b \), we get \( c=\frac{2}{3}b \).

Step1: Use the equation from 1a

We have the equation \( c=\frac{2}{3}b \) where \( c \) is pounds of clay and \( b \) is number of bowls.

Step2: Substitute \( b = 8 \) into the equation

Substitute \( b = 8 \) into \( c=\frac{2}{3}b \), so \( c=\frac{2}{3}\times8 \).

Step3: Calculate the value

\( \frac{2}{3}\times8=\frac{16}{3}=5\frac{1}{3} \). So we use the proportional relationship equation and plug in the number of bowls (8) to find the clay needed.

Step1: Define variables and relationship

Let the width of a rectangle be \( w \) (in cm) and the length be \( l \) (in cm). The relationship is \( l=w + 5 \) (since length is 5 cm longer than width).

Step2: Recall proportional relationship definition

A proportional relationship is of the form \( l=k\times w \) (where \( k \) is a constant, meaning the ratio \( \frac{l}{w} \) is constant for all \( w \)).

Step3: Analyze the ratio \( \frac{l}{w} \)

For \( l = w + 5 \), \( \frac{l}{w}=\frac{w + 5}{w}=1+\frac{5}{w} \). As \( w \) changes, \( \frac{5}{w} \) changes, so \( \frac{l}{w} \) is not constant. For example, if \( w = 1 \), \( l=6 \), \( \frac{l}{w}=6 \); if \( w = 2 \), \( l = 7 \), \( \frac{l}{w}=\frac{7}{2}=3.5 \), which are different. So the relationship is not proportional.

Answer:

\( c = \frac{2}{3}b \)

1b