COMPARING PROPORTIONAL RELATIONSHIPS

Problem 1 :

Meera is researching cruising speeds of different planes. Which airplane has a greater cruising speed ?

Solution :

Constant of proportionality from table :

k = y/x

k = 40/5

k = 8

From the table, the plane is covering 8 km per min.

Constant of proportionality from graph :

Point from the graph, say (1, 250)

k = 250/1

k = 250

From the graph, the plane is covering 250 km per min.

So, Jet Airliner is faster.

Problem 2 :

The distance covered by the fastest high speed train in Japan travelling at maximum speed is represented on the graph. The fastest high speed train in the United States Travelling at maximum speed covers 600 kilometers in 2 1/2 hours. Which train has a greater maximum speed ?

Solution :

Speed of train in US :

600 km in 2 1/2 hours

600 km --> 5/2 hours

1 hour --> 600 / (5/2)

1 hour ---> 240 km

Speed of train in Japan :

Choosing the point (1, 300) from the graph, we get

distance covered by the train in 1 hour = 300 km

So, speed of Japan's train is faster.

Problem 3 :

Amanda babysits and Petra does yard work on weekends. The graph relating Amanda's earnings to the number of hours babysits passes through the points (0, 0) and (4, 24). The table below relates Petra's earnings to the number of hours she doe yard work.

Who earns more per hour ?

Solution :

Amanda's earning :

From the given point (4, 24)

Constant of proportionality (k) = 24/4

k = 6

Amanda is earning $6 per hour from baby sitting.

Petra's earning :

(3, 15)

k = 15/3

k = 5

Petra is earning $5 per hour. 

So, Amanda earns more.

Problem 4 :

Milo pays $3 per pound for dog food at Pat's pet palace. The graph given below represents the cost per pound of food at Mark's Mutt market. At which store will Milo pay a lower price per pound for dog food ?

Solution :

Cost of dog food at Pat's pet palace = $3

From the given graph, (1.2, 5)

Constant of proportionality (k) = 5/1.2

= 4.16

Cost at Pat's pet palace is cheaper.

Problem 5 :

The graph represents the rate at which Marlo makes origami birds for a craft air. The equation y = 2.5x represents the number of birds, y Josh makes x in minutes. Who makes birds at a faster rate ?

Solution :

Constant of proportionality from graph :

One of the points from the line (10, 2)

k = 2/10

k = 1/5

y = 2.5x

Constant of proportionality from the equation :

y = kx

y = 2.5x

k = 2.5

k = 25/10

k = 5/2

So, Josh's rate is more.

Problem 6 :

Sam and Bobby want to know who cycled faster. The table shows the total miles Sam travelled over time. The graph shows the same relationship for Bobby. Who cycled faster ?

Solution :

Unit rate of Sam :

From the table choosing one of the points, say (2, 20).

k = 20/2

k = 10

Unit rate of Bobby :

Choosing one of the points from the line shown in the graph (2, 18).

k = 18/2

k = 9

Then, Bobby cycled faster.

Problem 7 :

The equation y = 15x can be used to determine the amount of money, y, Pauli's Pizzeria makes by selling x pizzas. The graph shows the money Leo's Pizzeria takes in for different number of pizzas sold. Which Pizzeria makes more money per pizza ?

Solution :

From the equation y = 15x

Unit rate of Pauli's Pizzeria = 15

Choosing one of the points from the graph, say (2, 24)

Unit rate of Leos Pizza = 24/2

k = 12

So, Pauli's Pizzeria will make more money.

Problem 8 :

The graph shows the amount of savings over time in Eliana's account. Lana, meanwhile puts $50 each week into her savings account. If they both begin with $0, who is saving greater rate ?

Solution :

From the given graph, (2, 94)

Unit rate of Eliana = 94/2

= 47

So, every month Eliana saves $47.

Lana saves $50 every month. 

So, Lana saves more.

Problem 9 :

Arshton, Alexa and Clara want to know who types the fastest. The equation y = 39x models the rate at which Arshton can type. Where y is the number of words typed and x is in time in minutes.

The table shows the relationship between words typed and minutes of Alexa.

The graph shows the same relationship for Clara. Who types the fastest ?

Solution :

Arshton :

y = 39x

Unit rate of Arshton = 39

Alexa :

(2, 78)

Unit rate of Alexa = 78/2

= 39

Clara :

From the graph, choosing one of the points, say (2, 78)

Unit rate of Clara = 78/2

= 39

All of the are typing in the same speed.