Question

homework 3: proportional relationships
directions: determine whether the quantities in each table represent a proportional relationship. if yes, give the constant rate.
1.
wages
hours pay
1 $9.50
2 $19.00
3 $28.50
4 $38.00
2.
gallons miles driven
2 50
3 72
4 96
5 125
3.
rain accumulation
hours inches
4 1.2
9 2.7
10 3
15 4.5
4.
green paint mix (oz)
blue yellow
5 7.5
12 18
18 27
24 36
5.
flour (c) sugar (c)
\\(\frac{1}{2}\\) 1
\\(1\frac{3}{4}\\) \\(3\frac{1}{2}\\)
2 4
\\(2\frac{1}{3}\\) \\(4\frac{2}{3}\\)
6.
customer service calls
calls duration (min)
3 27
4 35
5 48
6 54
directions: complete the table given each proportional relationship.

1. kiera pays $1.95 per song she downloads.

songs total cost
1
2
3
4

1. a bake shop uses \\(1\frac{1}{3}\\) cup of milk for each batch of cupcakes.

batches milk (c)
1
2
3
4

1. alec averaged 62 miles per hour on a recent road trip.

hours distance (mi)
1
4
7
9

1. grapes are on sale for $2.36 per pound.

pounds total cost
0.75
1.25
2
2.5

1. a plant is growing at a rate of 0.8 inches per week.

weeks height (in)
2
4
6
8

1. each quilt requires \\(3\frac{7}{8}\\) yards of fabric.

quilts fabric (yds)
1
2
3
4

Explanation:

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Tables 1-6: Proportionality Check

Step1: Define proportionality rule

Two quantities \(x\) and \(y\) are proportional if \(\frac{y}{x} = k\) (constant) for all pairs.

Step2: Check Table 1 (Wages)

Calculate \(\frac{\text{Pay}}{\text{Hours}}\):
$\frac{9.50}{1}=9.50$, $\frac{19.00}{2}=9.50$, $\frac{28.50}{3}=9.50$, $\frac{38.00}{4}=9.50$

Step3: Check Table 2 (Miles Driven)

Calculate \(\frac{\text{Miles Driven}}{\text{Gallons}}\):
$\frac{50}{2}=25$, $\frac{72}{3}=24$, $\frac{96}{4}=24$, $\frac{125}{5}=25$

Step4: Check Table 3 (Rain Accumulation)

Calculate \(\frac{\text{Inches}}{\text{Hours}}\):
$\frac{1.2}{4}=0.3$, $\frac{2.7}{9}=0.3$, $\frac{3}{10}=0.3$, $\frac{4.5}{15}=0.3$

Step5: Check Table 4 (Paint Mix)

Calculate \(\frac{\text{Yellow}}{\text{Blue}}\):
$\frac{7.5}{5}=1.5$, $\frac{18}{12}=1.5$, $\frac{27}{18}=1.5$, $\frac{36}{24}=1.5$

Step6: Check Table 5 (Flour & Sugar)

Calculate \(\frac{\text{Sugar}}{\text{Flour}}\):
$\frac{1}{1/2}=2$, $\frac{3\frac{1}{2}}{1\frac{3}{4}}=2$, $\frac{4}{2}=2$, $\frac{4\frac{2}{3}}{2\frac{1}{3}}=2$

Step7: Check Table 6 (Service Calls)

Calculate \(\frac{\text{Duration}}{\text{Calls}}\):
$\frac{27}{3}=9$, $\frac{35}{4}=8.75$, $\frac{48}{5}=9.6$, $\frac{54}{6}=9$
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Tables 7-12: Complete Proportional Tables

Step8: Fill Table 7 (Song Cost)

Total Cost = \(1.95 \times \text{Songs}\):
$1.95 \times 1 = 1.95$, $1.95 \times 2 = 3.90$, $1.95 \times 3 = 5.85$, $1.95 \times 4 = 7.80$

Step9: Fill Table 8 (Milk for Cupcakes)

Milk = \(1\frac{1}{3} \times \text{Batches} = \frac{4}{3} \times \text{Batches}\):
$\frac{4}{3} \times 1 = \frac{4}{3}=1\frac{1}{3}$, $\frac{4}{3} \times 2 = \frac{8}{3}=2\frac{2}{3}$, $\frac{4}{3} \times 3 = 4$, $\frac{4}{3} \times 4 = \frac{16}{3}=5\frac{1}{3}$

Step10: Fill Table 9 (Road Trip Distance)

Distance = \(62 \times \text{Hours}\):
$62 \times 1 = 62$, $62 \times 4 = 248$, $62 \times 7 = 434$, $62 \times 9 = 558$

Step11: Fill Table 10 (Grape Cost)

Total Cost = \(2.36 \times \text{Pounds}\):
$2.36 \times 0.75 = 1.77$, $2.36 \times 1.25 = 2.95$, $2.36 \times 2 = 4.72$, $2.36 \times 2.5 = 5.90$

Step12: Fill Table 11 (Plant Growth)

Height = \(0.8 \times \text{Weeks}\):
$0.8 \times 2 = 1.6$, $0.8 \times 4 = 3.2$, $0.8 \times 6 = 4.8$, $0.8 \times 8 = 6.4$

Step13: Fill Table 12 (Quilt Fabric)

Fabric = \(3\frac{7}{8} \times \text{Quilts} = \frac{31}{8} \times \text{Quilts}\):
$\frac{31}{8} \times 1 = 3\frac{7}{8}$, $\frac{31}{8} \times 2 = \frac{31}{4}=7\frac{3}{4}$, $\frac{31}{8} \times 3 = \frac{93}{8}=11\frac{5}{8}$, $\frac{31}{8} \times 4 = 31$

Answer:

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Proportionality Results (Tables 1-6)

1. Proportional, constant rate = $\$9.50$ per hour
2. Not proportional
3. Proportional, constant rate = $0.3$ inches per hour
4. Proportional, constant rate = $1.5$ oz yellow per oz blue
5. Proportional, constant rate = $2$ cups sugar per cup flour
6. Not proportional

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Completed Tables (Tables 7-12)

1. Songs & Total Cost:

• 1 song: $\$1.95$
• 2 songs: $\$3.90$
• 3 songs: $\$5.85$
• 4 songs: $\$7.80$

1. Batches & Milk:

• 1 batch: $1\frac{1}{3}$ c
• 2 batches: $2\frac{2}{3}$ c
• 3 batches: $4$ c
• 4 batches: $5\frac{1}{3}$ c

1. Hours & Distance:

• 1 hour: 62 mi
• 4 hours: 248 mi
• 7 hours: 434 mi
• 9 hours: 558 mi

1. Pounds & Total Cost:

• 0.75 lb: $\$1.77$
• 1.25 lb: $\$2.95$
• 2 lb: $\$4.72$
• 2.5 lb: $\$5.90$

1. Weeks & Height:

• 2 weeks: 1.6 in
• 4 weeks: 3.2 in
• 6 weeks: 4.8 in
• 8 weeks: 6.4 in

1. Quilts & Fabric:

• 1 quilt: $3\frac{7}{8}$ yds
• 2 quilts: $7\frac{3}{4}$ yds
• 3 quilts: $11\frac{5}{8}$ yds
• 4 quilts: 31 yds