Exploring Ratios, Equivalent Ratios, and Graphing Ratio Tables

	Fraction Form	Greatest Common Factor	Rewrite	Simplify
Tearrik	$\frac{2 7}{1 5}$	$GCF (27, 15) = 3$	$\frac{9 \cdot 3}{5 \cdot 3}$	$\frac{9}{5}$
Zain	$\frac{4 5}{2 5}$	$GCF (45, 25) = 5$	$\frac{9 \cdot 5}{5 \cdot 5}$	$\frac{9}{5}$

Fraction Form

Greatest Common Factor

Rewrite

Simplify

Tearrik

\frac{2 7}{1 5}

GCF (27, 15) = 3

\frac{9 \cdot 3}{5 \cdot 3}

\frac{9}{5}

Zain

\frac{4 5}{2 5}

GCF (45, 25) = 5

\frac{9 \cdot 5}{5 \cdot 5}

\frac{9}{5}

$\frac{1 4}{6}$
Scaling Back	Scaling Forward
$\frac{1 4 \div 2}{6 \div 2}$	$\frac{7 \times 3}{3 \times 3}$
$\frac{7}{3}$	$\frac{2 1}{9}$

\frac{1 4}{6}

Scaling Back

Scaling Forward

\frac{1 4 \div 2}{6 \div 2}

\frac{7 \times 3}{3 \times 3}

\frac{7}{3}

\frac{2 1}{9}

Distance Run (Miles)	$1$	$2$	$3$	$4$
Time (Minutes)	$10$	$20$	$30$	$40$

Distance Run (Miles)

1

2

3

4

Time (Minutes)

10

20

30

40

Height	$2$	$A$	$6$	$18$
Width	$3$	$6$	$9$	$B$

Height

2

A

6

18

Width

3

6

9

B

Height	$2$	$A$	$6$	$18$
Width	$3$	$6$	$9$	$B$

Height

2

A

6

18

Width

3

6

9

B

Height, $x$	$2$	$4$	$6$	$18$
Width, $y$	$3$	$6$	$9$	$27$

Height,

x

2

4

6

18

Width,

y

3

6

9

27

Maya	Mark
Days	Photos	Days	Photos
$1$	$5$	$1$	$7$
$2$	$10$	$2$	$14$
$3$	$15$	$3$	$21$
$4$	$20$	$4$	$28$
$5$	$25$	$5$	$35$

Maya

Mark

Days

Photos

Days

Photos

1

5

1

7

2

10

2

14

3

15

3

21

4

20

4

28

5

25

5

35

Maya
Days	Photos	$(x, y)$
$1$	$5$	$(1, 5)$
$2$	$10$	$(2, 10)$
$3$	$15$	$(3, 15)$
$4$	$20$	$(4, 20)$
$5$	$25$	$(5, 25)$

Maya

Days

Photos

(x, y)

1

5

(1, 5)

2

10

(2, 10)

3

15

(3, 15)

4

20

(4, 20)

5

25

(5, 25)

Mark
Days	Photos	$(x, y)$
$1$	$7$	$(1, 7)$
$2$	$14$	$(2, 14)$
$3$	$21$	$(3, 21)$
$4$	$28$	$(4, 28)$
$5$	$35$	$(5, 35)$

Mark

Days

Photos

(x, y)

1

7

(1, 7)

2

14

(2, 14)

3

21

(3, 21)

4

28

(4, 28)

5

35

(5, 35)

	Maya	Mark
Day, $x$	$1$	$1$
Photos, $y$	$5$	$7$

Maya

Mark

Day,

x

1

1

Photos,

y

5

7

Statement	True?
The ratio of photos to days for Maya is $5 : 1 .$	$✓$
The ratio of photos to days for Mark is $7 : 1 .$	$✓$
The line for Mark's data is steeper than the line for Maya's.	$✓$
The ratio of days to photos for Maya is $5 : 1 .$	$\times$
The ratio of days to photos for Mark is $7 : 1 .$	$\times$
The line for Maya's data is steeper than the line for Mark's.	$\times$

Statement

True?

The ratio of photos to days for Maya is

5 : 1 .

✓

The ratio of photos to days for Mark is

7 : 1 .

✓

The line for Mark's data is steeper than the line for Maya's.

✓

The ratio of days to photos for Maya is

5 : 1 .

\times

The ratio of days to photos for Mark is

7 : 1 .

\times

The line for Maya's data is steeper than the line for Mark's.

\times

Photos	$4$	$8$	$12$
Time (minutes)	$5$	$10$	$15$

Photos

4

8

12

Time (minutes)

5

10

15

Photos	$20$	$24$	$28$
Time (minutes)	$25$	$30$	$35$

Photos

20

24

28

Time (minutes)

25

30

35

Kevin can knit six sweaters in $10$ days.

How many sweaters can Kevin knit in

15

days?

If Kevin knits

12

sweaters, how many days does it take him to complete them?

We want to determine how many sweaters Kevin can knit in 15 days. We know that he can knit 6 sweaters in 10 days, so let's use this information to create a ratio table that relates the number of sweaters knit to the amount of time it takes to knit them.

Sweaters	6
Time	10

We can scale the ratio of 6 sweaters in 10 days, 6:10, back by dividing both quantities by 2. This will tell us how many sweaters Kevin can knit in 5 days.

Sweaters	6	6÷2=3
Time	10	10÷2=5

We found an equivalent ratio of 3 sweaters knit in 5 days. We can use this ratio to find how many sweaters Kevin can knit in 15 days. Since 5* 3=15, we can scale the 3:5 ratio forward by 3 to find the missing information.

Sweaters	6	6÷2=3	3* 3=9
Time	10	10÷2=5	5* 3=15

We found that Kevin can knit 9 sweaters in 15 days.

We want to determine how many days it takes Kevin to knit 12 sweaters, given that he can knit 6 sweaters in 10 days. Let's use this information to write the ratio of time spent knitting to the number of sweaters knit. 10days/6sweaters We know that 6* 2=12. This means we can multiply both quantities in this ratio by 2 to get an equivalent ratio with a denominator of 12 sweaters. 10days* 2/6sweaters* 2=20days/12sweaters We got 20 days in the numerator. This means that it took Kevin 20 days to knit 12 sweaters.

A vending machine at a school dispenses $70$ snacks in $5$ minutes.

How many snacks will the vending machine dispense in

7

minutes?

How long will it take for the vending machine to dispense

420

snacks?

We can create a ratio table to determine the number of snacks the vending machine would dispense in 7 minutes. We are given that the vending machine dispenses 70 snacks in 5 minutes. Let's add this information to the ratio table.

Snacks	70
Time	5

Note that there is no integer we can multiply 5 by to get 7. Let's first find the unit rate of snacks to time by scaling the ratio 70:5 back. We can do this by dividing both quantities in the ratio by 5 and recording the result in the second column of the ratio table. Remember, a unit ratio must have a denominator of 1.

Snacks	70	70÷5=14
Time	5	5÷5=1

Next, we can scale the unit ratio forward to 7 minutes by multiplying it by 7. This will give us the number of snacks the vending machine would dispense in 7 minutes.

Number of Snacks	70	14	14*7=98
Time	5	1	1*7=7

Therefore, the vending machine would dispense 98 snacks in 7 minutes.

Let's start by recalling that the vending machine dispenses 70 snacks in 5 minutes. We can represent this information as a ratio of time to snacks dispensed. 5minutes/70snacks We can use this ratio to create a ratio table that will help us to determine how long it would take for the vending machine to dispense 420 snacks. Let's multiply the numerator and the denominator of the ratio by 1, 2, 3, and so on, up to 7, to get the corresponding time and snacks for each equivalent ratio.

Time	5	10	15	20	25	30	35
Snacks	70	140	210	280	350	420	490

This means that it would take the machine 30 minutes to dispense 420 snacks. In this case, we were able to find the solution because 420 is a multiple of 70. Another number of snacks might require numbers beyond integers.

Dylan and Emily are putting together a charity fundraiser at their school. They have both been selling tickets for the event. Dylan has sold $3$ out of every $5$ tickets.

At the end of the day, Dylan realized he had sold

5

more tickets than Emily. How many tickets have they sold altogether?

Dylan and Emily sold tickets for a charity fundraiser event. We know that Dylan sold 3 out of every 5 tickets sold. Let's represent this information as the ratio of tickets sold by Dylan to the total number of tickets sold. 3:5 We want to determine the total number of tickets sold if Dylan sold five more tickets than Emily. First, let's create a ratio table with ratios that are equivalent to 3:5.

Tickets Sold by Dylan	3	6	9	12	15	18
Total Tickets Sold	5	10	15	20	25	30

We can find the number of tickets sold by Emily by calculating the difference between the total number of tickets sold and the number of tickets sold by Dylan.

Total Tickets Sold	Tickets Sold by Dylan	Difference	Tickets Sold by Emily
5	3	5-3	2
10	6	10-6	4
15	9	15-9	6
20	12	20-12	8
25	15	25-15	10
30	18	30-18	12

We now have the total number of tickets sold by both Dylan and Emily. Let's subtract the number of tickets sold by Emily from the number of tickets sold by Dylan to find the difference between the total tickets sold by Dylan and Emily.

Tickets Sold by Dylan	Tickets Sold by Emily	Difference	Extra Tickets Sold by Dylan
3	2	3-2	1
6	4	6-4	2
9	6	9-6	3
12	8	12-8	4
15	10	15-10	5
18	12	18-12	6

Now, we were told that that Dylan sold 5 tickets more than Emily. The only option in the table that meets this description is if 25 total tickets were sold.

A delivery truck travels $220$ miles in $4$ hours, while a delivery van travels $180$ miles in $3$ hours.

Which vehicle has a higher average speed?

How much farther can the faster vehicle travel in

5

hours?

Let's write the rate of the distance each vehicle travels to the amount of time it spent traveling. We know that the delivery truck travels 220 miles in 4 hours and that the delivery van travels 180 miles in 3 hours. cc Delivery Truck & & Delivery Van 220mi:4h & & 180mi:3h Next, we can determine the unit rates for each vehicle. Remember, finding the unit rate means that we need to get 1 on the right-hand side of each rate. Let's divide the delivery truck's rate by 4 to find its unit rate.

This means that the delivery truck travels 55 miles in 1 hour. Now let's calculate the unit rate for the delivery van by dividing both quantities by 3.

The delivery van travels 60 miles in 1 hour. Since the delivery van has a higher unit rate, it is faster than the delivery truck.

Consider the unit rates found in Part A for each vehicle. cc Delivery Truck & & Delivery Van 55mi:1h & & 60mi:1h We can create two equivalent rates by scaling up these rates by 5. That will show us the distance traveled by both vehicles after 5 hours. cc Delivery Truck& &Delivery Van 55mi* 5:1h* 5& &60mi* 5:1h* 5 ⇓& &⇓ 275mi:5h& &300mi:5h The delivery truck will travel 275 miles after 5 hours, while the delivery van will travel 300 miles. Let's subtract the distance the truck can travel from the distance the van can travel to find out how much further the delivery van will go. 300-275=25miles

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Comparing the Simplest Forms of a Pair of Ratios

Equivalent Ratios

Scaling Ratios

Determining if Two Ratios Are Equivalent

Mark's Favorite Shot

Hint

Solution

Ratio Table

Exploring the Aspect Ratio of Mark's New Phone Camera

Hint

Solution

Mark and Maya's Five-Day Photo Challenge

Hint

Solution

How Much Money Will Mark Earn?

Hint

Solution

Graphing Ratio Tables on a Coordinate Plane

Ratio Tables and Graphs

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