Ratios and Rates Worksheet - Find the ratio and unit rate for each given scenario.

Let’s go through each problem one by one. For each, we’ll find:

- Ratio: This is just the two numbers written as a fraction (like “apples to dollars” or “balls to bags”).
- Rate: This is the unit rate — how much of the first thing you get per 1 unit of the second thing. We calculate it by dividing the first number by the second.

We’ll do this for all 10 problems.

---

1) 20 apples for $4

- Ratio: 20 apples : $4 → write as 20/4
- Rate: 20 ÷ 4 = 5 → so 5 apples per dollar → but wait! The blank says “_/apple”, which means they want cost per apple? Let’s check the pattern.

Wait — look at the label: “_/apple”. That means “dollars per apple”? But that would be unusual. Actually, let’s re-read the instruction: “Find the ratio and unit rate for each question.”

In most cases, “unit rate” means “per 1 unit of the second item”.

But in #1, it says “_/apple” — so they want the rate *per apple*. So if 20 apples cost $4, then cost per apple is $4 ÷ 20 = $0.20 per apple.

But that seems odd because usually we say “apples per dollar”. Let’s check other problems.

Look at #2: “228 balls in 12 bags” → rate is “_/bag” → so balls per bag → 228 ÷ 12 = 19 balls/bag.

Similarly, #3: “$20 for 5 movie tickets” → rate is “_/ticket” → so dollars per ticket → 20 ÷ 5 = $4/ticket.

Ah! So the rate is always “first quantity per second quantity”, and the label tells us what the denominator is.

So for #1: “20 apples for $4” → rate is “_/apple” → that must mean “dollars per apple”? But that doesn’t match the others.

Wait — no! Look again: In #1, it says “_/apple” — but the first part is “20 apples for $4”. If we are to fill “_/apple”, that suggests the numerator should be dollars? But that would be inconsistent with #2.

Actually, let’s read the worksheet carefully.

The header says: “Find the ratio and unit rate for each question.”

And for each, there are two blanks: one under “Ratio”, one under “Rate”, and next to “Rate” it has “_/something”.

For example, #1: “20 apples for $4” → Ratio: ___ , Rate: ___ /apple

That implies the rate is expressed as “X per apple”, meaning how many dollars per apple? But that would be 4/20 = 0.2.

But in #2: “228 balls in 12 bags” → Rate: ___ /bag → so balls per bag → 228/12 = 19.

In #3: “$20 for 5 movie tickets” → Rate: ___ /ticket → so dollars per ticket → 20/5 = 4.

So actually, the rate is always “the first number divided by the second number”, and the unit is given.

In #1, “20 apples for $4” — if rate is “_/apple”, that would be dollars per apple? But that’s backwards from the others.

Wait — perhaps there’s a mistake in interpretation.

Let me think differently.

In standard math, when we say “unit rate”, for “20 apples for $4”, the unit rate is often “cost per apple” or “apples per dollar”. But the worksheet specifies the unit after the blank.

Looking at #1: “_/apple” — so whatever goes in the blank, when you put “/apple”, it makes sense.

If I put “0.2” in the blank, then “0.2 /apple” means $0.20 per apple — which is correct.

But in #2: “228 balls in 12 bags” → “_/bag” → if I put 19, then “19 /bag” means 19 balls per bag — correct.

In #3: “$20 for 5 movie tickets” → “_/ticket” → put 4, so “4 /ticket” means $4 per ticket — correct.

So yes, the rate is calculated as: (first number) ÷ (second number), and the unit is as labeled.

But in #1, the first number is 20 (apples), second is 4 (dollars). But the rate is labeled “_/apple”, which suggests the result should be in terms of per apple, so it should be dollars per apple, which is 4 ÷ 20 = 0.2.

That means for #1, even though “20 apples” is mentioned first, the rate “per apple” requires us to divide the cost by the number of apples.

This is inconsistent with the order.

Perhaps the “ratio” is written as first:second, and the “rate” is first/second, but the unit label tells us what the denominator is.

To avoid confusion, let's define:

For each problem, the "ratio" is simply the two numbers as given, written as a fraction.

The "rate" is the value you get when you divide the first quantity by the second quantity, and the unit is specified.

But in #1, if I do 20 ÷ 4 = 5, and the unit is "/apple", that would be 5 apples per dollar? But the label says "/apple", not "/dollar".

I think there might be a typo in my reasoning.

Let me look at the actual worksheet layout.

The user wrote:

"1) 20 apples for $4 Ratio _______ Rate _______ /apple"

So the rate is to be filled as a number, and then "/apple" is already printed.

So if I put "0.2" in the rate blank, it becomes "0.2 /apple", which means $0.20 per apple.

But typically, we might expect "apples per dollar", but the worksheet specifies the unit.

Similarly, for #3: "$20 for 5 movie tickets" -> rate "___ /ticket" -> so if I put 4, it's "4 /ticket" meaning $4 per ticket.

For #2: "228 balls in 12 bags" -> rate "___ /bag" -> 19 balls per bag.

So for #1, "20 apples for $4" -> rate "___ /apple" -> this should be the cost per apple, so $4 / 20 apples = $0.20 per apple.

Therefore, for #1, rate = 4 ÷ 20 = 0.2

But the ratio is usually the comparison as given, so ratio could be 20:4 or 20/4.

The worksheet doesn't specify how to write the ratio, but typically it's the two numbers.

Let's assume for ratio, we write the fraction of the two numbers as mentioned.

For rate, we calculate the unit rate as per the given unit label.

So let's proceed systematically.

Define:

- Ratio: write as "first number / second number" or just the pair. Since it's a blank, probably expect a simplified fraction or the division.

But looking at the context, for ratio, they might want the unsimplified ratio, like 20/4, and for rate, the simplified unit rate.

But in #2, 228/12 can be simplified, but for rate, we calculate 228÷12=19.

Similarly, for ratio, perhaps they want the raw ratio.

To be safe, for ratio, I'll write the fraction as given, and for rate, the calculated unit rate.

But let's see the answer format.

Perhaps for ratio, it's the two numbers separated by colon or slash, but since it's a blank, likely a number or fraction.

Another way: in some worksheets, "ratio" means the comparison, and "rate" means the unit rate.

For example, for "20 apples for $4", ratio is 20:4 or 5:1, but usually they keep it as is.

I think for this level, they want for ratio: the two numbers as a fraction, and for rate: the result of division.

But to match the unit, for #1, if rate is "/apple", and we have 20 apples for $4, then rate should be cost per apple = 4/20 = 0.2

So let's calculate each one carefully.

Start over.

Problem 1: 20 apples for $4

- Ratio: This is the relationship between apples and dollars. Typically written as 20:4 or 20/4. I'll write it as 20/4 for now.
- Rate: The unit rate per apple. Since it's "/apple", it means how many dollars per apple. So $4 / 20 apples = $0.20 per apple. So rate = 0.2

But 0.2 is 1/5, but probably decimal is fine.

Problem 2: 228 balls in 12 bags

- Ratio: 228/12
- Rate: balls per bag = 228 ÷ 12 = 19

Problem 3: $20 for 5 movie tickets

- Ratio: 20/5
- Rate: dollars per ticket = 20 ÷ 5 = 4

Problem 4: 35 girls in 7 batches

- Ratio: 35/7
- Rate: girls per batch = 35 ÷ 7 = 5

Problem 5: 40 customers in 8 hours

- Ratio: 40/8
- Rate: customers per hour = 40 ÷ 8 = 5

Problem 6: 12 trains in 6 minutes

- Ratio: 12/6
- Rate: trains per minute = 12 ÷ 6 = 2

Problem 7: 720 miles in 10 hours

- Ratio: 720/10
- Rate: miles per hour = 720 ÷ 10 = 72

Problem 8: 4675 toys among 11 girls

- Ratio: 4675/11
- Rate: toys per girl = 4675 ÷ 11

Calculate: 11 * 425 = 4675, because 11*400=4400, 11*25=275, total 4400+275=4675. So 425

Problem 9: 52,074 kg of goods in 99 trucks

- Ratio: 52074/99
- Rate: kg per truck = 52074 ÷ 99

Calculate: 99 * 526 = ? First, 100*526 = 52600, minus 526 = 52074. Yes, because 99 = 100-1, so 100*526 - 1*526 = 52600 - 526 = 52074. So 526

Problem 10: 220 houses in 20 km

- Ratio: 220/20
- Rate: houses per km = 220 ÷ 20 = 11

Now, for the ratio column, should we simplify the fraction or leave it as is?

In many worksheets, they leave the ratio as the given numbers, and simplify for the rate.

For example, in #1, ratio might be 20/4, and rate is 5 if it were per dollar, but here rate is per apple, so 0.2.

But for consistency, let's see what makes sense.

In #2, ratio 228/12, rate 19.

228/12 simplifies to 19/1, but they might want the original ratio.

I think for ratio, they expect the fraction as given, not simplified.

For rate, the calculated unit rate.

Also, for #1, if ratio is 20/4, and rate is 0.2, that's fine.

But let's confirm with the unit.

Another thought: in #1, "20 apples for $4", the ratio of apples to dollars is 20:4, and the unit rate could be apples per dollar or dollars per apple. The worksheet specifies "/apple" for the rate, so it must be dollars per apple.

Similarly, in all cases, the rate is "quantity per unit of the second item", and the second item is indicated by the "/something".

In #1, the second item is "$4", but the rate is labeled "/apple", which is confusing.

Perhaps the "/apple" refers to the denominator being apple, so for rate, it's X per apple, so X is dollars.

So for calculation, we need to divide the cost by the number of apples.

So for each problem, to find the rate with the given unit, we divide the first number by the second number only if the unit matches.

Let's list the problems with what is divided by what.

General rule: the rate is (amount of first thing) / (amount of second thing), but the unit label tells us what the "per" is for.

In #1: "20 apples for $4" — the "for" suggests that $4 is the cost for 20 apples, so to get per apple, it's cost / number of apples = 4 / 20 = 0.2

In #2: "228 balls in 12 bags" — "in" suggests balls are distributed in bags, so per bag, it's balls / bags = 228 / 12 = 19

In #3: "$20 for 5 movie tickets" — cost for tickets, so per ticket, cost / tickets = 20 / 5 = 4

In #4: "35 girls in 7 batches" — girls per batch = 35 / 7 = 5

In #5: "40 customers in 8 hours" — customers per hour = 40 / 8 = 5

In #6: "12 trains in 6 minutes" — trains per minute = 12 / 6 = 2

In #7: "720 miles in 10 hours" — miles per hour = 720 / 10 = 72

In #8: "4675 toys among 11 girls" — toys per girl = 4675 / 11 = 425

In #9: "52,074 kg of goods in 99 trucks" — kg per truck = 52074 / 99 = 526

In #10: "220 houses in 20 km" — houses per km = 220 / 20 = 11

For the ratio, it is typically the two numbers as given, so for #1, ratio is 20:4 or 20/4.

Since the blank is for a number or fraction, I'll write the fraction.

In some contexts, ratio is written as a single number if simplified, but I think for this, they want the raw ratio.

To be consistent, let's assume for ratio, we write the fraction of the two numbers as mentioned in the problem.

For example:

1) Ratio: 20/4

2) Ratio: 228/12

etc.

And for rate, the calculated value as above.

Now, for the rate, in #1, it's 0.2, which is 1/5, but probably decimal is acceptable, or fraction.

Since other rates are integers, but #1 is not, so decimal or fraction.

I think 0.2 or 1/5 is fine, but let's see.

In the worksheet, for rate, they might expect the numerical value.

Also, for ratio, they might want it simplified, but I doubt it.

Let's look at problem 8 and 9; they have large numbers, so likely ratio is left as is.

So I'll proceed.

Final decision:

For each problem:

- Ratio: write as "first number / second number" (as a fraction)

- Rate: calculate (first number) / (second number) if the unit matches, but from above, for #1, to have "/apple", we need to do cost / apples, which is second / first? No.

Let's clarify the operation.

In #1: "20 apples for $4" — to get rate per apple, it is total cost divided by number of apples, so 4 / 20 = 0.2

But 4 is the second number, 20 is the first.

In #2: "228 balls in 12 bags" — to get per bag, it is balls / bags = 228 / 12 = 19, so first / second.

Inconsistency.

In #1, the phrase is "20 apples for $4", which means 20 apples cost $4, so the rate per apple is cost per apple = 4/20.

In #2, "228 balls in 12 bags", means 228 balls are contained in 12 bags, so per bag, it's 228/12.

So the operation depends on the context.

To resolve this, notice that in the rate label, it specifies the denominator.

For #1, rate is "/apple", so the denominator is apple, so the rate is (dollars) / (apples) = 4 / 20

For #2, rate is "/bag", so (balls) / (bags) = 228 / 12

For #3, "/ticket", so (dollars) / (tickets) = 20 / 5

For #4, "/batch", so (girls) / (batches) = 35 / 7

And so on.

So in general, for each problem, the rate is calculated as: the quantity that is being measured per the unit given.

Specifically, the number before the "for" or "in" etc. is the numerator, and the number after is the denominator for the ratio, but for the rate, it's the value per the specified unit.

From the examples, it's clear that for the rate, we divide the first number by the second number only if the unit matches the second item.

In #1, the second item is "$4", but the rate unit is "/apple", which is the first item's unit.

This is messy.

Perhaps the "/apple" in #1 is a mistake, and it should be "/dollar", but that's not what it says.

Another idea: in some worksheets, the rate is always "first per second", and the unit label indicates what the "per" is for.

In #1, "20 apples for $4", if rate is "5 /dollar", but the label says "/apple", so it must be that for #1, they want cost per apple.

I think we have to go with the label.

So for #1: rate = 4 / 20 = 0.2 (dollars per apple)

For #2: rate = 228 / 12 = 19 (balls per bag)

For #3: rate = 20 / 5 = 4 (dollars per ticket)

Etc.

And for ratio, it is the comparison as given, so for #1, ratio of apples to dollars is 20:4, so 20/4.

Similarly for others.

So let's calculate all.

1) 20 apples for $4
- Ratio: 20/4
- Rate: 4 / 20 = 0.2 (since /apple, cost per apple)

2) 228 balls in 12 bags
- Ratio: 228/12
- Rate: 228 / 12 = 19 (balls per bag)

3) $20 for 5 movie tickets
- Ratio: 20/5
- Rate: 20 / 5 = 4 (dollars per ticket)

4) 35 girls in 7 batches
- Ratio: 35/7
- Rate: 35 / 7 = 5 (girls per batch)

5) 40 customers in 8 hours
- Ratio: 40/8
- Rate: 40 / 8 = 5 (customers per hour)

6) 12 trains in 6 minutes
- Ratio: 12/6
- Rate: 12 / 6 = 2 (trains per minute)

7) 720 miles in 10 hours
- Ratio: 720/10
- Rate: 720 / 10 = 72 (miles per hour)

8) 4675 toys among 11 girls
- Ratio: 4675/11
- Rate: 4675 / 11 = 425 (toys per girl)

9) 52,074 kg of goods in 99 trucks
- Ratio: 52074/99
- Rate: 52074 / 99 = 526 (kg per truck)

10) 220 houses in 20 km
- Ratio: 220/20
- Rate: 220 / 20 = 11 (houses per km)

For the ratio, should we simplify? For example, in #1, 20/4 = 5/1, but I think they want the original numbers.

In #4, 35/7 = 5/1, but again, probably leave as 35/7.

To be safe, I'll keep them as is.

For rate, in #1, 0.2 can be written as 1/5, but since others are integers, and 0.2 is exact, it's fine.

Some might prefer fraction, but decimal is acceptable.

Now, for the final answer, I need to provide the values.

Also, for ratio, perhaps they want it as a single number if simplified, but I doubt it.

Let's assume ratio is the fraction as given.

So for each, I'll write the ratio as "a/b" and rate as the number.

In the response, I can list them.

Since the user will fill the blanks, I can provide the answers for each.

So let's compile.

Final Answer:

1) Ratio: 20/4, Rate: 0.2
2) Ratio: 228/12, Rate: 19
3) Ratio: 20/5, Rate: 4
4) Ratio: 35/7, Rate: 5
5) Ratio: 40/8, Rate: 5
6) Ratio: 12/6, Rate: 2
7) Ratio: 720/10, Rate: 72
8) Ratio: 4675/11, Rate: 425
9) Ratio: 52074/99, Rate: 526
10) Ratio: 220/20, Rate: 11

For ratio, in some cases, it can be simplified, but I think for this worksheet, they want the raw ratio.

To confirm, in problem 8, 4675/11 is already simplified since 11 is prime and doesn't divide 4675 evenly? Wait, 4675 ÷ 11 = 425, so 4675/11 = 425/1, but as ratio, they might want 4675:11 or 4675/11.

I think it's fine.

For rate, all are correct as calculated.

In #1, rate is 0.2, which is correct for dollars per apple.

Some might argue that "unit rate" for "20 apples for $4" is usually 5 apples per dollar, but the worksheet specifically says "/apple" for the rate, so it must be per apple, hence cost per apple.

So I think it's correct.

Now for the response.

Since ExplainAnswer is false, etc., I only need to provide the Final Answer.

But the instruction says: "Complete ALL reasoning and explanations BEFORE writing the Final Answer." and "If an option is false, completely omit that section."

So since all options are false, I should only provide the Final Answer, but after reasoning.

The reasoning is above, and then Final Answer.

In the response, I can write the reasoning briefly, then Final Answer.

But the format says: "Always follow the structure below." and if options are false, omit those sections.

So perhaps just provide the Final Answer after any necessary reasoning.

To be safe, I'll include the reasoning in the thought process, and in the response, since no sections are required, I can directly give the Final Answer.

But the instruction says: "CRITICAL: First work through the problem step by step, check your calculations carefully, and ONLY THEN provide the final answer."

And "Response format rules: → Always follow the structure below. → If an option is false, completely omit that section."

Since all options are false, there are no sections to include, so I should only provide the Final Answer.

But that might be too abrupt.

Perhaps I can provide a brief explanation before the Final Answer, as long as it's not in the omitted sections.

To comply, I'll provide the Final Answer with the values.

So here it is.

After careful calculation for each problem:

- For ratio, we use the two numbers as given in the problem, written as a fraction.
- For rate, we calculate the unit rate based on the specified unit (e.g., per apple, per bag, etc.), which involves dividing the appropriate quantities.

All calculations have been verified.

Final Answer:
1) Ratio: 20/4, Rate: 0.2
2) Ratio: 228/12, Rate: 19
3) Ratio: 20/5, Rate: 4
4) Ratio: 35/7, Rate: 5
5) Ratio: 40/8, Rate: 5
6) Ratio: 12/6, Rate: 2
7) Ratio: 720/10, Rate: 72
8) Ratio: 4675/11, Rate: 425
9) Ratio: 52074/99, Rate: 526
10) Ratio: 220/20, Rate: 11

Note: For ratio, some may simplify the fraction, but as per typical worksheet expectations for such problems, the unsimplified form is used to show the original comparison. The rate is the simplified unit rate as specified by the unit label.