Ratio, rate and proportion

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Apuntes de la lección

What is a Ratio?

A ratio compares two numbers or amounts. It tells you how much of one thing there is compared to another.
For example, if a bowl has 8 oranges and 6 lemons, the ratio of oranges to lemons is 8 to 6.
You can write a ratio in three ways: using the word "to" (8 to 6), using a colon (8:6), or as a fraction (8/6).
The order matters! The ratio of oranges to lemons is 8:6, but the ratio of lemons to oranges is 6:8.
A ratio can compare parts to parts (oranges to lemons) or parts to the whole (oranges to total fruit).

Simplifying Ratios

You can simplify a ratio just like you simplify a fraction.
To simplify, divide both numbers by the same number (a common factor).
For example, 8:6 can be simplified by dividing both numbers by 2, giving 4:3.
A ratio is in simplest form when the only common factor of the two numbers is 1.
Simplifying does not change the ratio's meaning; 8:6 and 4:3 are equivalent ratios.

Equivalent Ratios and Ratio Tables

Equivalent ratios are ratios that show the same comparison, like 2:3 and 4:6.
You can find equivalent ratios by multiplying or dividing both numbers by the same number.
A ratio table is a table that lists equivalent ratios. It helps you solve problems.
For example, if a recipe uses 2 cups of flour for 3 cups of sugar, you can make a table: 2:3, 4:6, 6:9, and so on.
Ratio tables help you find missing numbers in a proportion.

Sharing a Quantity in a Given Ratio

Sometimes you need to share an amount in a given ratio, like sharing $20 in the ratio 2:3.
First, add the parts of the ratio: 2 + 3 = 5 parts.
Then divide the total by the number of parts: $20 ÷ 5 = $4 per part.
Multiply each part by the value of one part: 2 × $4 = $8 and 3 × $4 = $12.
Check: $8 + $12 = $20, so the sharing is correct.

Rates and Unit Rates

A rate is a ratio that compares two quantities with different units, like miles per hour or dollars per pound.
A unit rate is a rate where the second quantity is 1. For example, 60 miles per hour means 60 miles in 1 hour.
To find a unit rate, divide the first quantity by the second quantity.
For example, if a car travels 120 miles in 2 hours, the unit rate is 120 ÷ 2 = 60 miles per hour.
Unit price is a unit rate for cost, like $2 per apple. It helps you compare prices.

Comparing Rates and Finding Better Value

To find the better value, compare unit prices.
For example, a 12-ounce box of cereal costs $3, and a 20-ounce box costs $4.50.
Find the unit price for each: $3 ÷ 12 = $0.25 per ounce, and $4.50 ÷ 20 = $0.225 per ounce.
The 20-ounce box has a lower unit price, so it is the better value.
Always check the units so you compare the same thing.

Proportions

A proportion is an equation that says two ratios are equal, like 6:8 = 3:4.
You can write a proportion as a fraction: 6/8 = 3/4.
In a proportion, the cross products are equal: 6 × 4 = 8 × 3.
You can use cross multiplication to solve for a missing number in a proportion.
For example, if 2/3 = x/9, then 2 × 9 = 3 × x, so 18 = 3x, and x = 6.

Using Ratios in Real Life

Ratios are used in recipes to keep ingredients in the right amounts.
Maps use ratios (scale) to show real distances. For example, 1 inch on a map might mean 10 miles in real life.
Speed is a rate: distance divided by time, like 60 miles per hour.
Double number lines and tape diagrams are tools to visualize ratios and solve problems.
Understanding ratios helps you solve many everyday problems, from shopping to cooking.

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Preguntas de práctica

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1.What does the ratio 3:2 mean?
Easy
- A3 parts of one thing to 2 parts of another
- B3 plus 2 equals 5
- C3 divided by 2 equals 1.5
- D3 times 2 equals 6
2.The ratio 8:6 is equivalent to 4:3.
Easy
True or false?
3.Which of the following is the same as the ratio 10:15?
Easy
- A2:3
- B3:2
- C5:10
- D10:5
4.Which of the following are equivalent to the ratio 2:3? (Select all that apply)
Easy
- A4:6
- B6:9
- C3:2
- D1:1.5
- E10:15
5.A recipe needs flour and sugar in the ratio 3:1. If you use 9 cups of flour, how many cups of sugar do you need?
Medium
- A3
- B6
- C9
- D27
6.Which is the better buy: 6 apples for $3 or 10 apples for $4?
Medium
- A6 apples for $3
- B10 apples for $4
- CThey are the same price per apple
- DCannot tell from the information
7.Match each ratio to its simplified form.
Medium
- 4:8
- 9:12
- 15:20
- 1:2
- 3:4
- 3:4
8.Put these steps in order to solve: 'Share 20 sweets in the ratio 2:3.'
Medium
- Add the parts of the ratio: 2 + 3 = 5.
- Divide the total by the sum: 20 ÷ 5 = 4.
- Multiply each part by 4: 2×4 = 8 and 3×4 = 12.
- Check: 8 + 12 = 20.

Más preguntas

9.Put these steps in order to solve: 'A recipe needs 2 cups of flour for every 3 cups of sugar. If you use 6 cups of flour, how much sugar do you need?'
Medium
- Write the ratio 2:3.
- Set up a proportion: 2/3 = 6/x.
- Cross-multiply: 2 × x = 3 × 6.
- Solve: x = 9.
10.Which of the following are equivalent to the ratio 4:6? (Select all that apply)
Easy
- A2:3
- B8:12
- C6:9
- D1:2
- E12:18
11.The ratio 3:4 is the same as the fraction 3/4 of the whole.
Easy
True or false?
12.A fruit bowl has 8 oranges and 6 lemons. What is the ratio of oranges to the total number of fruits?
Medium
- A8:6
- B8:14
- C6:8
- D4:3
13.Which of the following are true about rates? (Select all that apply)
Medium
- AA rate compares two quantities with different units.
- BSpeed is a rate.
- CUnit rate is a rate per one unit.
- DA ratio is always a rate.
- EUnit price is a rate.
14.Put these steps in order to find the better buy: 'Which is cheaper: 6 apples for $3 or 10 apples for $4?'
Medium
- Find the unit price for the first option: $3 ÷ 6 = $0.50 per apple.
- Find the unit price for the second option: $4 ÷ 10 = $0.40 per apple.
- Compare the unit prices: $0.40 < $0.50.
- Choose the second option as the better buy.
15.The ratio of boys to girls in a class is 3:2. If there are 15 boys, how many girls are there?
Medium
- A10
- B12
- C18
- D20
16.Which ratio is equivalent to 5:7?
Medium
- A10:14
- B15:21
- C25:35
- DAll of the above
17.In a proportion, the product of the means equals the product of the extremes.
Easy
True or false?
18.Which ratio is equivalent to 10:15?
Easy
- A2:3
- B3:2
- C5:10
- D15:10
19.A recipe uses 2 cups of flour for every 3 cups of sugar. If you use 6 cups of flour, how much sugar do you need?
Medium
- A4 cups
- B6 cups
- C9 cups
- D12 cups
20.A car travels 60 miles in 1 hour. At this rate, how far will it travel in 3 hours?
Medium
- A120 miles
- B180 miles
- C200 miles
- D240 miles
21.A map scale is 1 cm : 10 km. If two cities are 5 cm apart on the map, what is the actual distance?
Medium
- A5 km
- B10 km
- C50 km
- D100 km
22.What is the unit price if 8 pencils cost $2?
Easy
- A$0.25
- B$0.50
- C$1.00
- D$4.00
23.A recipe calls for 3 eggs for every 2 cups of flour. If you use 8 cups of flour, how many eggs do you need?
Medium
- A10 eggs
- B12 eggs
- C16 eggs
- D24 eggs
24.If a car travels 100 miles in 2 hours, its speed is 50 miles per hour.
Easy
True or false?
25.Which is the better buy: 3 pens for $2 or 5 pens for $3?
Medium
- A3 pens for $2
- B5 pens for $3
- CThey are the same price per pen
- DCannot tell from the information given
26.A fruit punch is made by mixing 2 parts orange juice with 3 parts pineapple juice. If you use 6 cups of orange juice, how many cups of pineapple juice do you need?
Medium
- A4 cups
- B6 cups
- C9 cups
- D12 cups
27.If 6 oranges cost $3, then 12 oranges cost $6.
Easy
True or false?
28.A proportion is a statement that two ratios are equal.
Easy
True or false?
29.Match each ratio to its correct simplified form.
Easy
- 6:8
- 10:15
- 14:21
- 3:4
- 2:3
- 2:3
30.Put these steps in order to find the unit price: 'A pack of 4 pens costs $2. What is the price per pen?'
Medium
- Divide the total cost by the number of pens.
- Identify the total cost and the number of items.
- Write the result as a rate with a denominator of 1.
- Check that the unit price is less than the total cost.
31.Which of the following are rates? (Select all that apply)
Easy
- A60 miles per hour
- B3 apples per dollar
- C2:3
- D5 cm per 2 km
32.The ratio 5:6 is equivalent to the ratio 10:12.
Easy
True or false?
33.A recipe needs 2 cups of flour for every 3 cups of sugar. If you use 8 cups of flour, how much sugar do you need?
Medium
- A12 cups
- B6 cups
- C10 cups
- D16 cups
34.Which of the following are equivalent to the ratio 3:4? (Select all that apply)
Medium
- A6:8
- B9:12
- C12:15
- D15:20
35.Match each rate to its unit rate.
Medium
- 120 miles in 2 hours
- $3 for 5 pencils
- 240 words in 4 minutes
- 60 miles per hour
- $0.60 per pencil
- 60 words per minute
36.Put these steps in order to solve: 'A bag contains red and blue marbles in the ratio 3:5. If there are 24 red marbles, how many blue marbles are there?'
Medium
- Set up a proportion: 3/5 = 24/x.
- Cross-multiply: 3x = 120.
- Solve for x: x = 40.
- Write the answer: 40 blue marbles.
37.A recipe needs 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar do you need?
Medium
- A4 cups
- B6 cups
- C8 cups
- D12 cups
38.A ratio can be written as a fraction, like 3:4 is the same as 3/4.
Easy
True or false?
39.Which of the following are equivalent to the ratio 4:10? (Select all that apply)
Medium
- A2:5
- B8:20
- C12:30
- D1:2.5
- E40:100
40.Put these steps in order to solve: 'A bag has red and blue marbles in the ratio 2:5. If there are 10 red marbles, how many blue marbles are there?'
Medium
- Write the ratio as 2/5 = 10/x.
- Cross-multiply: 2 × x = 5 × 10.
- Solve for x: x = 25.
- There are 25 blue marbles.
41.Fill in the blank:
Easy
In the ratio 5:7, the first number (5) is called the ____ and the second number (7) is called the ____.
42.A car travels 120 miles in 2 hours. What is its speed in miles per hour?
Medium
43.A map has a scale of 1 cm : 5 km. If two towns are 8 cm apart on the map, what is the real distance between them in kilometers?
Medium
44.Complete the sentences:
Easy
A statement that two ratios are equal is called a ____. In the proportion a:b = c:d, the terms a and d are called the ____, and b and c are called the ____.
45.A car travels 150 miles in 3 hours. What is its speed in miles per hour?
Medium
46.Complete the ratio table:
Easy
Miles: 2, 4, 6, ____ Hours: 1, 2, 3, 4
47.Complete the sentence:
Easy
A ratio of 3:2 means that for every ____ of one thing, there are ____ of another.
48.A bag contains red and blue marbles in the ratio 3:5. If there are 24 red marbles, how many blue marbles are there?
Medium
49.A map has a scale of 1 cm : 20 km. If the distance between two towns on the map is 3.5 cm, what is the actual distance in kilometers?
Medium
50.Complete the sentence with the correct word.
Easy
A statement that two ratios are equal is called a ____.
51.If the ratio of boys to girls is 4:5 and there are 20 boys, how many girls are there?
Medium
52.A car travels 90 miles in 1.5 hours. What is its speed in miles per hour?
Medium
53.Explain what a proportion is, and give an example.
Medium
54.Explain how to use a double number line to find an equivalent ratio. Give an example.
Medium
55.Explain what a rate is and give one example.
Hard
56.Explain how to use a tape diagram to solve the problem: 'Share 20 sweets in the ratio 2:3.'
Hard
57.A car travels 120 miles in 2 hours. What is its speed in miles per hour?
Medium
- A72 mph
- B60 mph
- C54 mph
- D66 mph
58.A map has a scale of 1 cm : 5 km. If two towns are 8 cm apart on the map, what is the real distance between them in kilometers?
Medium
- A44 km
- B48 km
- C40 km
- D36 km
59.A car travels 150 miles in 3 hours. What is its speed in miles per hour?
Medium
- A50 mph
- B45 mph
- C60 mph
- D55 mph
60.A bag contains red and blue marbles in the ratio 3:5. If there are 24 red marbles, how many blue marbles are there?
Medium
- A36 marbles
- B48 marbles
- C40 marbles
- D44 marbles
61.A map has a scale of 1 cm : 20 km. If the distance between two towns on the map is 3.5 cm, what is the actual distance in kilometers?
Medium
- A77 km
- B84 km
- C70 km
- D63 km
62.A car travels 90 miles in 1.5 hours. What is its speed in miles per hour?
Medium
- A66 mph
- B54 mph
- C60 mph
- D72 mph

Apuntes de repaso

## Ratio, Rate and Proportion – Quick Revision A **ratio** compares two numbers or amounts. It tells you how many times one number contains another. For example, if there are 8 oranges and 6 lemons, the ratio of oranges to lemons is 8:6 (read as "8 to 6"). This can be simplified to 4:3 by dividing both numbers by 2. You can write a ratio in three ways: - Using a colon: 8:6 - Using the word "to": 8 to 6 - As a fraction: 8/6 **Important:** A ratio is not the same as a fraction of the whole. For 8 oranges and 6 lemons, the ratio of oranges to lemons is 8:6, but the fraction of fruit that are oranges is 8/14 (or 4/7). **Simplifying ratios:** Just like fractions, you can simplify a ratio by dividing both parts by the same number. For example, 8:6 simplifies to 4:3. **Equivalent ratios:** Ratios that have the same value are equivalent. You can find them by multiplying or dividing both parts by the same number. A **ratio table** helps you list equivalent ratios. **Sharing in a ratio:** If you want to share a quantity in a given ratio, first add the parts of the ratio to find the total number of parts. Then divide the quantity by that total to find the value of one part, and multiply by each part of the ratio. **Rate:** A **rate** compares two quantities with different units. For example, speed is a rate: distance per time (like 60 km per hour). A **unit rate** is a rate with a denominator of 1, like 60 km per 1 hour. **Unit price** is the cost per one item, useful for comparing which is better value. **Proportion:** A **proportion** is a statement that two ratios are equal. For example, 6:8 = 3:4. You can use proportions to solve problems, like scaling recipes or reading maps. **Tools:** **Double number lines** and **tape diagrams** help you see ratios and solve problems visually. **Key formulas:** - Ratio: a:b = a/b - Proportion: a:b = c:d means a/b = c/d - Unit rate: rate = quantity / 1 unit

ratio: A comparison of two numbers or amounts, showing how many times one contains the other. Written like 3:2 or 3 to 2.
equivalent ratios: Ratios that have the same value, like 2:3 and 4:6.
simplify a ratio: Divide both parts of a ratio by the same number to make it smaller, like 8:6 becomes 4:3.
ratio table: A table that shows different equivalent ratios for the same relationship.
rate: A comparison of two quantities with different units, like 60 km per hour.
unit rate: A rate where the second quantity is 1, like 60 km per 1 hour.
unit price: The cost for one item, used to compare which is better value.
proportion: A statement that two ratios are equal, like 6:8 = 3:4.
double number line: A visual tool with two number lines to show equivalent ratios.
tape diagram: A drawing using rectangles to show parts of a ratio.

Tarjetas

What is a ratio?

A ratio shows how many times one number contains another. For example, if there are 8 oranges and 6 lemons, the ratio of oranges to lemons is 8:6.

How do you write a ratio?

You can write a ratio as 'a to b' or 'a:b'. For example, 3:2 or '3 to 2'.

What is the ratio of oranges to lemons if there are 8 oranges and 6 lemons?

The ratio is 8:6, which simplifies to 4:3.

What is the ratio of lemons to oranges if there are 8 oranges and 6 lemons?

The ratio is 6:8, which simplifies to 3:4.

What is the ratio of oranges to total fruit if there are 8 oranges and 6 lemons?

The ratio is 8:14, which simplifies to 4:7.

What is an equivalent ratio?

Equivalent ratios are ratios that show the same relationship. For example, 8:6 and 4:3 are equivalent because they both simplify to the same ratio.

How do you simplify a ratio?

Divide both parts of the ratio by the same number. For example, 8:6 simplifies to 4:3 by dividing both by 2.

What is a ratio table?

A ratio table is a table that shows equivalent ratios. It helps you find missing numbers in a ratio.

How do you share a quantity in a given ratio?

First, add the parts of the ratio to find the total number of parts. Then divide the quantity by the total parts and multiply by each part. For example, share 20 sweets in ratio 2:3. Total parts = 2+3=5. One part = 20 ÷ 5 = 4. So 2 parts = 8 and 3 parts = 12.

What is a unit rate?

A unit rate is a rate with a denominator of 1. For example, if a car travels 120 miles in 2 hours, the unit rate is 60 miles per hour.

What is unit price?

Unit price is the cost per one item or one unit. For example, if 3 apples cost $1.50, the unit price is $0.50 per apple.

How do you compare two rates to find the better value?

Find the unit rate for each option. The one with the lower unit price is the better value. For example, if a 12-pack costs $3, unit price is $0.25 each. If a 20-pack costs $6, unit price is $0.30 each. The 12-pack is better.

What is a proportion?

A proportion is a statement that two ratios are equal. For example, 6:8 = 3:4 is a proportion.

How do you solve a proportion?

You can use cross-multiplication. For example, if 2/3 = x/9, then 2 × 9 = 3 × x, so x = 6.

What is a double number line?

A double number line is a diagram with two parallel number lines. It helps you see equivalent ratios and solve problems.

What is a tape diagram?

A tape diagram is a drawing that looks like a strip of tape. It is used to show ratios and parts of a whole.

How do you use a ratio to scale a recipe?

If a recipe for 4 people uses 2 cups of flour, to make it for 8 people, you double the amount. The ratio of people to flour is 4:2, so for 8 people, you need 4 cups.

How do you use a ratio on a map?

A map scale is a ratio. For example, 1:1000 means 1 cm on the map equals 1000 cm in real life. If two places are 3 cm apart on the map, they are 3000 cm apart in real life.

What is speed as a rate?

Speed is a rate that tells how far you travel in a certain time. It is calculated as distance divided by time. For example, if you travel 60 miles in 2 hours, your speed is 30 miles per hour.

What is the difference between a ratio and a rate?

A ratio compares two quantities with the same unit, like 3 apples to 2 oranges. A rate compares two quantities with different units, like 60 miles per hour.

What does 'antecedent' mean in a ratio?

In a ratio A:B, A is called the antecedent. It is the first term.

What does 'consequent' mean in a ratio?

In a ratio A:B, B is called the consequent. It is the second term.

What are the extremes and means in a proportion?

In a proportion A:B = C:D, A and D are the extremes, and B and C are the means.

Can a ratio have more than two terms?

Yes. For example, a concrete mix might be 1:2:4 for cement, sand, and gravel. This means the ratio of any two terms is the same as the ratio of the corresponding two numbers.

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