Lesson 5 Homework Practice Proportional And Nonproportional ... - Free Printable

Let’s go through each problem one by one. I’ll explain what to do and how to find the answer — step by step, like we’re solving it together.

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Problem 1: Fluffy Weighted

We are given:
- Days: 0 → 25 lb, 7 → 34 lb, 14 → 43 lb, 21 → 52 lb

To check if proportional, we look at the rate of change (how much weight per day).

From day 0 to 7:
Weight gain = 34 - 25 = 9 lb in 7 days → rate = 9/7 ≈ 1.286 lb/day

From day 7 to 14:
43 - 34 = 9 lb in 7 days → same rate = 9/7

From day 14 to 21:
52 - 43 = 9 lb in 7 days → again 9/7

So the rate is constant! That means it’s proportional.

Constant of proportionality = 9/7 (since every 7 days, gains 9 lbs)

✔ Answer: Yes, proportional. Constant = 9/7

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Problem 2: Miles Driven

Given:
- Gallons: 0 → 0 mi, 1 → 26 mi, 2 → 52 mi, 3 → 78 mi, 4 → 104 mi

Check rates:

From 0 to 1 gal: 26 - 0 = 26 mi/gal
From 1 to 2 gal: 52 - 26 = 26 mi/gal
From 2 to 3 gal: 78 - 52 = 26 mi/gal
From 3 to 4 gal: 104 - 78 = 26 mi/gal

All rates equal → proportional!

Constant of proportionality = 26 miles per gallon

✔ Answer: Yes, proportional. Constant = 26

*(Note: The original says “No; the rates are not equal” — that’s wrong. Let me double-check... Wait, maybe they meant something else? But looking at numbers: 26, 52, 78, 104 — all multiples of 26. So definitely proportional. Maybe a typo in the worksheet? We go with math.)*

Actually — wait! Look again: when gallons = 0, miles = 0 → good start for proportionality. And each step adds 26. So yes, proportional.

But the worksheet says “No”. Hmm. Let me re-read...

Wait — maybe they think because 0→0 isn’t counted? No, that’s fine. Or perhaps misread the table?

No — based on data, it IS proportional. I think the worksheet has an error here. But since you’re doing homework, let’s see what they expect.

Wait — actually, looking back: the worksheet says “No; the rates are not equal.” But our calculation shows they ARE equal.

Hold on — maybe I misread the table? Let me copy again:

Gallons: 0, 1, 2, 3, 4
Miles: 0, 26, 52, 78, 104

Yes — difference between consecutive miles: always +26. So rate is constant.

Therefore, it SHOULD be “Yes”.

But since the worksheet says “No”, maybe there’s a trick? Or maybe it’s a mistake.

I think we should trust the math. So I’ll say:

✔ Correct Answer: Yes, proportional. Constant = 26

*(If your teacher marked it as “No”, ask them why — because mathematically, it is proportional.)*

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Problem 3: Shad working job

Shad works 15 hours a week.

Question: Does this show constant of proportionality? Then write equation relating hours worked to weeks. How many days to work total of 120 hours? Use 15c = 3 weeks.

First part: Is it proportional?

He works 15 hours per week → so hours = 15 × weeks → YES, proportional.

Equation: h = 15w (h = hours, w = weeks)

Now: He wants to work 120 hours total.

Use 15c = 3 weeks → this seems confusing. Probably typo.

It says: “Use 15c = 3 weeks” — maybe meant “use 15 hours per week”?

Assume: 15 hours per week → so to get 120 hours:

120 ÷ 15 = 8 weeks

But question asks: “How many days will it take?”

Assuming he works 5 days per week? Not specified.

Wait — it says “Use 15c = 3 weeks” — maybe c is days? Unclear.

Perhaps “15c = 3 weeks” means 15 hours per week, and 3 weeks is example?

Better interpretation: Since he works 15 hours/week, then for 120 hours:

Number of weeks = 120 / 15 = 8 weeks

If we assume 5 school/work days per week → 8 × 5 = 40 days

But the problem doesn’t specify days per week.

Wait — it says “Use 15c = 3 weeks” — maybe c is number of days? Like 15 hours over c days equals 3 weeks? Doesn’t make sense.

Alternative: Perhaps “15c = 3 weeks” is meant to be “15 hours per week”, and “c” is a variable.

Maybe it’s saying: if 15 hours = 1 week, then 120 hours = ? weeks → 8 weeks.

Then convert to days: if 1 week = 7 days → 8 × 7 = 56 days? But that’s calendar days, not work days.

This is messy.

Looking at context: probably expects us to use 15 hours per week, and find weeks first, then maybe assume 5 days/week.

But let’s read carefully: “Use 15c = 3 weeks”

Perhaps it’s a hint: 15 hours corresponds to 3 weeks? That would mean 5 hours per week? But earlier said 15 hours a week.

Contradiction.

Wait — original: “Shad started a new job working 15 hours a week.”

Then: “Use 15c = 3 weeks”

Maybe “c” is the constant? Or typo.

Another idea: perhaps “15c = 3 weeks” means that 15 hours takes 3 weeks? But that contradicts “15 hours a week”.

I think there’s a typo in the problem.

Best guess: ignore “Use 15c = 3 weeks” as confusing, and go with standard.

He works 15 hours per week.

Total hours needed: 120

Weeks needed: 120 / 15 = 8 weeks

If we assume he works 5 days per week → 8 × 5 = 40 days

If 7 days → 56 days

But typically in such problems, unless specified, we might assume 5-day work week.

However, the problem says “how many days”, and gives “Use 15c = 3 weeks” — perhaps c is days per week?

Suppose 15 hours = 3 weeks → then hourly rate? No.

Let’s try: if 15 hours = 3 weeks, then 1 week = 5 hours? But that contradicts “15 hours a week”.

I think the “Use 15c = 3 weeks” is likely a mistake or misprint.

Perhaps it’s “use 15 hours per week”, and “c” is weeks.

I’ll proceed with:

Hours = 15 × weeks

For 120 hours: weeks = 120 / 15 = 8

Now, to find days: if no info, perhaps assume 1 week = 7 days? But that’s unusual for jobs.

In many textbooks, they assume 5-day work week unless stated.

But let’s see the answer format — probably expects a number.

Another thought: “Use 15c = 3 weeks” might mean that c is the number of days, and 15 hours over c days equals 3 weeks of work? Still unclear.

Perhaps “15c = 3 weeks” is meant to be solved for c, but 3 weeks is time, 15c is hours? Units don't match.

I think safest is to calculate weeks first: 8 weeks.

Then, since the problem mentions “days”, and no specification, perhaps they want total days assuming continuous work? Unlikely.

Wait — in the initial statement: “working 15 hours a week” — implies weekly schedule.

Perhaps “Use 15c = 3 weeks” is irrelevant or typo.

I recall that in some contexts, “c” might stand for something, but here it's ambiguous.

Let me look for similar problems online or standard approach.

Standard way: if someone works H hours per week, to work T hours, weeks = T/H.

Then days = weeks × days_per_week.

Since not specified, and to match common practice, I'll assume 5 days per week.

So 8 weeks × 5 days/week = 40 days.

If the "15c = 3 weeks" is meant to indicate that 15 hours correspond to 3 weeks, then rate is 5 hours per week, but that contradicts "15 hours a week".

Unless "15 hours a week" is wrong, but it's stated clearly.

I think there's a typo, and "Use 15c = 3 weeks" should be ignored or is "use 15 hours per week".

Perhaps "c" is the constant, but in proportionality, constant is 15 hours/week.

I'll go with 40 days, assuming 5-day work week.

But let's calculate exactly as per text.

Another interpretation: "Use 15c = 3 weeks" might mean that for c days, he works 15 hours, and that equals 3 weeks of work? But 3 weeks of what?

This is too vague.

Perhaps "15c = 3 weeks" is an equation to solve for c, but 3 weeks is not a number.

I think the best course is to state:

Since he works 15 hours per week, to work 120 hours, he needs 120 / 15 = 8 weeks.

If we assume a 5-day work week, then 8 * 5 = 40 days.

If the problem intends something else, it's poorly worded.

But in many similar problems, they expect 40 days.

Perhaps "Use 15c = 3 weeks" means that 15 hours is done in c days, and that period is 3 weeks, so c = 21 days for 15 hours? Then rate is 15 hours / 21 days = 5/7 hours per day.

Then for 120 hours: days = 120 / (5/7) = 120 * 7/5 = 168 days.

That seems too long, and contradicts "15 hours a week" — if he works 15 hours in 21 days, that's less than 1 hour per day, while "15 hours a week" suggests about 3 hours per day if 5 days.

Inconsistent.

I think the "Use 15c = 3 weeks" is likely a distractor or typo, and we should use the clear information: 15 hours per week.

So I'll go with 8 weeks, and for days, since not specified, perhaps the problem expects just the weeks, but it asks for days.

Another idea: perhaps "c" is the number of days per week, and "15c = 3 weeks" is not literal.

I give up on that part. Let's move to other problems and come back.

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Problem 4: Sunflower growth

Grows 4 cm/day. After 28 days, height is 212 cm.

Find constant of proportionality. Equation. Height after 40 days.

First, is it proportional? If it grows at constant rate from day 0, then yes.

But after 28 days, height is 212 cm. If it started at 0, then rate = 212 / 28 = ? Let's calculate.

212 ÷ 28 = 7.571... but it says grows 4 cm/day. Contradiction?

Read: "During its first 28 days of growth, a sunflower grows about 4 centimeters a day. Find the constant of proportionality. Then write an equation relating height in centimeters to days."

But if it grows 4 cm/day, then after 28 days, height should be 4 * 28 = 112 cm, but it says 212 cm. Inconsistency.

Perhaps the 4 cm/day is the rate, but it didn't start from 0? But for proportionality, it should start from 0.

The problem says "grows about 4 centimeters a day", and then gives actual height after 28 days as 212 cm, which is not 112, so either the rate is not 4, or it's not proportional from 0.

But it asks for constant of proportionality, implying it is proportional.

Perhaps the 4 cm/day is approximate, and we should use the actual data.

Let's see: after 28 days, 212 cm. If proportional, constant k = height / days = 212 / 28.

Calculate: 212 ÷ 28 = 53/7 ≈ 7.571 cm/day.

But the problem says "grows about 4 cm/day" — that must be a red herring or mistake.

Perhaps "grows about 4 cm/day" is for a different plant, but no, it's for this sunflower.

Another possibility: the 4 cm/day is the average, but we have exact data.

I think for the purpose of this problem, since it gives specific height at 28 days, and asks for constant of proportionality, we should use that.

So, if proportional, h = k * d

At d=28, h=212, so k = 212 / 28 = 53/7 ≈ 7.571 cm/day

Simplify 212/28: divide numerator and denominator by 4: 53/7

So k = 53/7

Equation: h = (53/7) d

Then after 40 days: h = (53/7) * 40 = (53 * 40) / 7 = 2120 / 7 ≈ 302.857 cm

But the problem says "grows about 4 cm/day", which is confusing.

Perhaps the 4 cm/day is incorrect, or perhaps it's the initial rate, but changes.

But the problem asks for constant of proportionality, so likely assumes constant rate from 0.

And the 212 cm at 28 days is the data point.

So I'll go with that.

Constant = 212/28 = 53/7 cm/day

Equation: h = (53/7)d

After 40 days: h = (53/7)*40 = 2120/7 = 302 6/7 cm or approximately 302.86 cm

But let's keep as fraction: 2120/7 cm

Or simplify: 2120 ÷ 7 = 302.857..., but better as fraction.

2120 and 7, 7 is prime, 2120 ÷ 7 not integer, so 2120/7 cm.

But perhaps reduce: 2120/7 is already simplified.

Note that 212/28 = 53/7, and 53/7 * 40 = 2120/7.

Yes.

But the "4 cm/day" is puzzling. Perhaps it's a typo, and it's supposed to be consistent.

Maybe "grows about 4 cm/day" is for comparison, but we use the given height.

I think for accuracy, use the data given.

So constant = 212/28 = 53/7 cm/day

Equation: h = (53/7)d

Height at 40 days: (53/7)*40 = 2120/7 cm

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Problem 5: Number of Messages

Table:
Minutes: 5, 10, 15, 20, 25, 30
Messages: 50, 40, 30, 20, 10, 0

Determine pattern, form proportion, identify constant.

First, as minutes increase, messages decrease.

From 5 to 10 min: messages from 50 to 40, decrease of 10
10 to 15: 40 to 30, decrease 10
15 to 20: 30 to 20, decrease 10
etc.

So for every 5 minutes, messages decrease by 10.

So rate of change = -10 messages / 5 minutes = -2 messages per minute.

But is it proportional? Proportional usually means y = kx, passing through origin.

Here, when minutes=0, what is messages? From pattern, at min=0, messages=60? Because at min=5, 50, so if linear, intercept is 60.

Let's see: at min=0, if extrapolate, messages=60.

At min=30, messages=0.

So it's linear, but not proportional because when x=0, y≠0.

Proportional requires y=0 when x=0.

Here, when minutes=0, messages=60 (assumed), not 0.

So not proportional.

The relationship is linear: m = -2t + b

At t=5, m=50: 50 = -2*5 + b => 50 = -10 + b => b=60

So m = -2t + 60

Not proportional.

Constant of proportionality doesn't apply since not proportional.

The worksheet says "Yes: 0.1" — that doesn't make sense.

0.1 what? Messages per minute? But it's decreasing.

Perhaps they mean the slope magnitude, but still.

Or perhaps they think it's proportional with negative constant, but usually proportionality implies direct variation through origin.

In this case, it's not through origin.

For example, at t=0, m=60 ≠0.

So not proportional.

The worksheet answer "Yes: 0.1" is likely wrong.

Perhaps they calculated something else.

Another thought: maybe "constant of proportionality" for the rate of change, but typically for proportional relationships, it's k in y=kx.

Here, it's affine, not proportional.

So I think correct answer is: not proportional.

But let's see the values: the ratio messages/minutes: at t=5, 50/5=10; t=10, 40/10=4; t=15, 30/15=2; not constant, so not proportional.

Whereas for proportional, ratio should be constant.

Here, ratios are 10,4,2,1,0.4,0 — not constant.

So definitely not proportional.

Worksheet says "Yes: 0.1" — perhaps they meant the slope is -2, and |slope| =2, not 0.1.

Or perhaps they calculated minutes/messages or something.

At t=30, m=0, undefined.

I think worksheet has error.

Correct: not proportional.

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Problem 6: Kitchen faucet

Water runs out at 1.5 gallons per minute.

Complete table.

Minutes: 0.75, 1.5, 1.875, 2.25, 3

Subtraction of water: ?

Rate is 1.5 gal/min, so amount subtracted = rate × time

So for each minute value, multiply by 1.5.

At 0.75 min: 1.5 * 0.75 = 1.125 gal

At 1.5 min: 1.5 * 1.5 = 2.25 gal

At 1.875 min: 1.5 * 1.875 = let's calculate: 1.5 * 1.875

1.5 * 1.875 = 1.5 * (15/8) = (3/2)*(15/8) = 45/16 = 2.8125 gal

Or 1.5 * 1.875: 1.5 * 1.8 = 2.7, 1.5 * 0.075 = 0.1125, total 2.8125

At 2.25 min: 1.5 * 2.25 = 1.5 * 9/4 = 13.5/4 = 3.375 gal

At 3 min: 1.5 * 3 = 4.5 gal

So table:

Minutes: 0.75, 1.5, 1.875, 2.25, 3

Subtraction: 1.125, 2.25, 2.8125, 3.375, 4.5

The worksheet has "mins 1.5" at the end, probably indicating the rate.

Constant of proportionality is 1.5 gallons per minute.

Relationship is proportional since at time 0, subtraction=0.

Equation: s = 1.5 t, where s is subtraction in gallons, t in minutes.

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Problem 7: Cooking turkey

Amount of time t to cook varies with weight w.

Recommended 20 minutes per pound.

Is cooking time proportional to weight?

Yes, because t = 20 * w

So constant of proportionality is 20 minutes per pound.

Sample answer says: "No; Sample answer: The ratio cooking time / weight is not the same for all weights."

But according to recommendation, it is the same: 20 min/lb for all weights.

So it should be yes.

Unless in reality it's not, but the problem says "it is recommended that a turkey be cooked for 20 minutes for each pound", so for the recommendation, it is proportional.

So sample answer is wrong.

Correct: yes, proportional, constant = 20 min/lb.

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Now, back to Problem 3.

Let me try to interpret "Use 15c = 3 weeks".

Perhaps "c" is the number of days, and "15c = 3 weeks" means that 15 hours is worked in c days, and that period is equivalent to 3 weeks of work, but that doesn't help.

Another idea: perhaps "15c = 3 weeks" is meant to be "15 hours per week", and "c" is a variable for weeks, but then "3 weeks" is example.

Or perhaps it's "for c weeks, he works 15c hours", and "3 weeks" is given as example, so for 3 weeks, 45 hours, but we need 120 hours.

So 15c = 120, c=8 weeks.

Then for days, if we assume 5 days per week, 40 days.

Perhaps "Use 15c = 3 weeks" means that the constant c is such that 15c = 3, so c=3/15=0.2, but what is c?

I think it's safe to assume that he works 15 hours per week, so for 120 hours, 8 weeks, and if we must give days, and no specification, perhaps the problem expects 8 weeks, but it asks for days.

In the context, "how many days will it take", and since he works weekly, perhaps they want the number of days he works, assuming 5 days per week.

So 8 weeks * 5 days/week = 40 days.

I'll go with that.

So for Problem 3:
- Yes, proportional (hours = 15 * weeks)
- Equation: h = 15w
- For 120 hours, w = 120/15 = 8 weeks
- Assuming 5 days per week, days = 8 * 5 = 40 days

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Now, to summarize answers:

1. Yes, proportional. Constant = 9/7

2. Yes, proportional. Constant = 26 (despite worksheet saying no)

3. Yes, proportional. Equation: h = 15w. For 120 hours, 8 weeks, which is 40 days (assuming 5-day work week)

4. Constant = 212/28 = 53/7 cm/day. Equation: h = (53/7)d. After 40 days, h = (53/7)*40 = 2120/7 cm

5. Not proportional (ratios not constant, and not through origin)

6. Subtraction: 1.125, 2.25, 2.8125, 3.375, 4.5 gallons. Constant = 1.5 gal/min

7. Yes, proportional. Constant = 20 min/lb

But for the final answer, since the user may expect the answers as per the worksheet or corrected, I'll provide the mathematically correct ones.

Also, for Problem 2, I think the worksheet has a mistake; it should be yes.

Similarly for others.

So let's box the answers as per calculation.

Final Answer:

1. Yes, proportional; constant = \frac{9}{7}

2. Yes, proportional; constant = 26

3. Yes, proportional; equation: h = 15w; for 120 hours, it takes 40 days (assuming 5-day work week)

4. Constant = \frac{53}{7} cm/day; equation: h = \frac{53}{7}d; height after 40 days = \frac{2120}{7} cm

5. Not proportional

6. Subtraction amounts: 1.125, 2.25, 2.8125, 3.375, 4.5 gallons; constant = 1.5

7. Yes, proportional; constant = 20

But to match the format, perhaps list them clearly.

Since the user may have the worksheet with answers, but I think we should correct errors.

For the response, I'll write the answers as above.

In the final answer section, put the key points.

Perhaps for each problem, state the answer.

But the instruction is to provide final answer after reasoning.

So I'll compile.