(3). The second picture represents proportional relationship.
(4). The proportional relationship with \(\dfrac{4}{5}\) is \(\boxed{\frac{{20}}{{25}}}\). Option (B) is correct.
Further explanation:
Explanation:
The points in the first picture are \(\left( {1,1} \right), \left( {3,4} \right),\left( {5,7} \right)\) and \(\left( {6,8.25} \right).\)
The slopes between the points can be obtained as follows,
\(\begin{aligned}m&=\frac{{4 - 1}}{{3 - 1}}\\&=\frac{3}{2}\\\end{aligned}\)
\(\begin{aligned}m&=\frac{{7 - 4}}{{5 - 3}}\\&= \frac{3}{2}\\\end{aligned}\)
\(\begin{aligned}m&= \frac{{8.25 - 7}}{{6 - 5}}\\&= \frac{{1.25}}{1}\\\end{aligned}\)
The slopes between the points are not equal. Therefore, in the first picture x and y are not in a proportional relationship.
The points in the second picture arec\(\left( {1,2} \right), \left( {2,4} \right),\left( {4,8} \right)\) and \(\left( {6,10} \right).\)
The slopes between the points can be obtained as follows,
\(\begin{aligned}m&= \frac{{4 - 2}}{{2 - 1}}\\&= 2\\\end{aligned}\)
\(\begin{aligned}m&= \frac{{8 - 4}}{{4 - 2}}\\&= 2\\\end{aligned}\)
\(\begin{aligned}m&=\frac{{12 - 8}}{{6 - 4}}\\&= 2\\\end{aligned}\)
The slopes between the points are equal. Therefore, in the second picture x and y are in a proportional relationship.
The points in the third picture are \(\left( {0,0} \right), \left( {2,5} \right),\left( {4,8} \right)\) and \(\left( {6,10} \right).\)
The slopes between the points can be obtained as follows,
\(\begin{aligned}m&= \frac{{5 - 0}}{{2 - 0}}\\&= \frac{5}{2}\\\end{aligned}\)
\(\begin{aligned}m&=\frac{{8 - 5}}{{4 - 2}}\\&=\frac{3}{2}\\\end{aligned}\)
The slopes between the points are not equal. Therefore, in the third picture x and y are not in a proportional relationship.
The points in the first picture are \(\left( {1,12} \right), \left( {2,10} \right),\left( {5,4} \right)\) and \(\left( {6,2} \right).\)
The slopes between the points can be obtained as follows,
\(\begin{aligned}m&=\frac{{10 - 12}}{{3 - 1}}\\&= \frac{{ - 2}}{1}\\\end{aligned}\)
\(\begin{aligned}m&= \frac{{4 - 10}}{{5 - 2}}\\&= - 2\\\end{aligned}\)
\(\begin{aligned}m&= \frac{{2 - 4}}{{6 - 5}}\\&= \frac{{ - 2}}{1}\\\end{aligned}\)
The slopes between the points are equal but the slope is negative. Therefore, in the fourth picture x and y are not in a proportional relationship.
Part (4)
The ratio of red candies to green candies in a bag is \(\dfrac{4}{5}.\)
In option (A)
The ratio can be calculated as follows,
\({\text{Ratio}} = \dfrac{{16}}{{25}}\)
In option (B)
The ratio can be calculated as follows,
\(\begin{aligned}{\text{Ratio}}&= \frac{{20}}{{25}}\\&=\frac{4}{5}\\\end{aligned}\)
In option (C)
The ratio can be calculated as follows,
\(\begin{aligned}{\text{Ratio}}&= \frac{{20}}{{35}}\\&= \frac{4}{7}\\\end{aligned}\)
In option (D)
The ratio can be calculated as follows,
\(\begin{aligned}{\text{Ratio}}&=\frac{{24}}{{32}}\\&= \frac{3}{4}\\\end{aligned}\)
The proportional relationship with \(\dfrac{4}{5}\) is \(\boxed{\frac{{20}}{{25}}}\). Option (B) is correct.
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Ratio and proportion
Keywords: function, First picture, second picture, proportional relationship, ratio, slope, green candies, bag, red candies, 4/5, same relation, set, set of values, set of numbers, coordinates, x-coordinate, y-coordinate.
