# Ratio and proportions worksheets: Proportions Worksheets — free & printable

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Finding Ratios

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Finding Ratios Visual

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Each worksheet has 11 problems finding the ratio of shapes.

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Ratio Wording

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Each worksheet has 15 problems interpreting the ratio described.

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Each worksheet has 15 problems using and finding rate terminology.

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Reducing Ratios

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Each worksheet has 20 problems reducing a ratio to its lowest form.

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Understanding Ratios (Word)

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Each worksheet has 10 word problems finding the ratio, other half of a ratio or total number in a ratio.

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Ratios Double Line

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Each worksheet has 8 problems using a double line graph to answer ratio questions.

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Writing Equations from Ratios

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Each worksheet has 15 problems expressing a measurement ratio as an equation.

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Using Ratio Equations

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Each worksheet has 12 problems using an equation to find an equivalent measurement.

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Creating Equivalent Ratios

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Each worksheet has 20 problems filling in the blank to generate an equivalent ratio.

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Determining Proportionality with Tables

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Each worksheet has 12 problems determining if the values in a table are proportional.

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Creating Examples for Ratios

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Each worksheet has 10 problems writing an example of a ratio.

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Identifying True and False Ratio Statements

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Each worksheet has 6 problems

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Finding Rate

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Using Unit Prices

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Each worksheet has 10 problems using unit rate to find the answer.

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Ratios and Unit Rates

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Each worksheet has 15 problems find the ratio and unit rate of a scenario.

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Finding Equivalent Unit Fraction with Fractions

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Each worksheet has 14 problems finding an equivalent unit fraction using fractions.

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Using Unit Rates with Fractions

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Each worksheet has 10 problems finding the unit rate with fractions.

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Understanding Unit Rate

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Each worksheet has 12 problems determining the solution to a problem with unit rates.

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Constant of Proportionality (tables)

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Each worksheet has 8 problems using a table to identify the constant of proportionality.

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Constant of Proportionality (Graphs)

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Each worksheet has 8 problems using a graph to identify the constant of proportionality.

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Examining Y=KX

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Each worksheet has 10 problems using the formula Y=KX to solve a word problem.

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Explaining X and Y with Proportionality

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Each worksheet has 6 problems determining the significance of x and y in relation to proportionality.

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Creating Tables and Graphs of Ratios

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Each worksheet has 4 problems creating a table and graph from a scenario.

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## Ratio and Proportion Worksheets | Definitions, Examples, Activities

Worksheets /Math /Ratio and Proportion Worksheets

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Table of Contents

Ratios are pairs of numbers that say how much of one thing compares to another. They can be written in three different ways: with the word *to*, with a *colon*, or as a *fraction*.

**See the fact file below for more information on Ratio and Proportions, or you can download our 22-page** **Ratio and Proportions worksheet pack to utilze within the classroom or home environment.**

### Key Facts & Information

#### RATIO AND PROPORTION

- The connection between the amounts of two or more items is defined by ratio. It is used to contrast portions of the same type.
- It is considered to be in proportion if two or more ratios are equal.
- There are two methods to display the proportion. It can be represented with either an equal sign or a colon symbol, as in a:b = c:d or a:b:: c:d.
- It does not affect the ratio whether we multiply or divide each term by the same number.
- If the ratio between the first and second amounts is identical to the ratio between the second and third quantities, the quantities are said to be in continuing proportion.

##### Ratio

- A ratio compares two quantities produced by dividing the first by the second. If a and b are two values of the same kind and with the same units, and b is not equal to zero, then the quotient a/b is known as the ratio between a and b.
- The colon sign is used to denote ratios (:). The ratio a/b has no unit and may be expressed as a: b.

##### Proportion

- The equality of two ratios is referred to as proportion. Two equal ratios are always proportional. Proportions are represented by the symbol (::) and assist us in solving for unknown values.
- The proportion is an equation or statement that shows that two ratios or fractions are comparable.
- If a: b = c: d, four non-zero values, a, b, c, and d, are said to be in proportion.
- Consider the ratios 10:20 and 250:500. In this case, 10:20 may be written as 10:20 = 10/20 = 0.5, and 250:500 can be written as 250:500 = 250/500 = 10/20 = 0.5. Because both ratios are equal, they are proportionate.
- Proportions are classified into three kinds.
- Direct Proportion
- Inverse Proportion
- Continuous Proportion

##### Direct Proportion

- The term “direct proportion” refers to the link between two quantities. When one amount rises, the other rises, and vice versa.
- As a result, a direct proportion is expressed as y∝x. For example, increasing the speed of an automobile causes it to travel more distance in a given amount of time.

##### Inverse Proportion

- Inverse proportion explains the connection between two quantities in which one rises while the other falls and vice versa.
- As a result, an inverse proportion is expressed as y∝1/x. For example, when a vehicle’s speed increases, it will travel a constant distance in less time.

##### Continuous Proportion

- This refers to the relationship between three or more quantities where the ratio between each two consecutive terms is always the same.
- To express the continuous proportion, you would not use the least common multiple of the means, but the common ratio, which is the ratio between the terms.
- For example, consider the following three ratios: a:b, b:c, and c:d. The common ratio is the ratio between consecutive terms, in this case, b:c. So the continuous proportion can be expressed as a:b:c:d or a:b=b:c=c:d.
- The LCM of b and c for the given ratio is not used here.
- It’s important to note that in continuous proportion, we are not multiplying any of the terms, we are just expressing the proportionality between the terms in consecutive order.

#### FORMULA

- For example, the ratio 3: 2 can be written as 3/2, where 3 is the antecedent and 2 is the consequent.
- To indicate a proportion for the two ratios, a: b and c: d, we write them as a:b::c:d⟶a/b=c/d
- The terms b and c are known as mean terms.
- The terms a and d are referred to as extreme terms.
- In a: b = c: d, the values a and b must be of the same kind and have the same units, but c and d may be of the same type and have the same units independently. For example: 4 kg: 20 kg = Rs. 60: Rs. 300.
- The proportion formula is written as a/b = c/d or a: b:: c: d.
- The product of the means equals the outcome of the extremes in proportion.
- As a result of the percentage formula a: b:: c: d, we obtain bxc = axd. For instance, in 4: 20: 60: 300, we get 4×20 = 60×300.

#### DIFFERENCE BETWEEN RATIO AND PROPORTION

- The following table shows the differences between ratio and proportion.

RATIO | PROPORTION |
---|---|

It is used to compare the size of two quantities with the same measurement unit. | It is used to describe the relationship between two ratios. |

A colon (:) and a slash (/) are used to denote a ratio. | A proportion is represented with a double colon (::) |

It is known as an expression. | It is known as an equation. |

- You can compare any two amounts with the same units.
- Only if two ratios are equal are they considered to be in proportion.
- We may also use the cross-product approach to determine whether two ratios are equal and in proportion.
- The ratio stays the same whether we multiply and divide each term by the same number.
- If the ratio between the first and second amounts is identical to the ratio between the second and third, the quantities are said to be in a continuous proportion.
- Similarly, the ratio between the first and second items in a continuous proportion equals the ratio between the third and fourth.

### Ratio and Proportion Worksheets

This is a fantastic bundle that includes everything you need to know about Ratio and Proportion across 22 in-depth pages. These are** ready-to-use worksheets that are perfect for teaching students about Ratio and Proportion, mathematical concepts that describe the relationship between two or more values.**

### Complete List Of Included Worksheets

- Math: Ratio and Proportion Facts
- Word Play
- Applications
- What is the R&P
- Equivalent Ratios
- Mixed Jumbled Numbers
- On and Off
- Alt Sign
- Cross Multiplication
- Proportion Types
- Problem Solving

#### Frequently Asked Questions

##### What is the difference between ratio and proportion?

A ratio compares two or more quantities expressed as fractions or decimals. For example, the ratio of apples to bananas in a basket can be expressed as 3:2, meaning there are 3 apples for every 2 bananas.

A proportion, however, is an equation that states that two ratios are equal. For example, if the ratio of apples to bananas in a basket is 3:2, and there are 6 apples, then the proportion of apples to bananas would be 3/2 = 6/x, where x is the number of bananas. Solving for x, we can find that x = 4, so there are 4 bananas in the basket.

##### How do you solve a proportion?

To solve a proportion, you can use cross-multiplication. Cross-multiplication involves multiplying both sides of the proportion by the same value to isolate the unknown variable. For example, to solve the proportion 3/2 = 6/x, you can cross-multiply to get 3 * x = 2 * 6, and then divide both sides by 3 to get x = 4.

##### What is the meaning of the symbol “:” in a ratio?

The symbol “:” is used to express a ratio. For example, the ratio of apples to bananas in a basket can be written as 3:2, meaning there are 3 apples for every 2 bananas. The symbol “:” is read as “to” or “per”.

##### Can a ratio be expressed as a decimal or a percentage?

Yes, a ratio can be expressed as a decimal or a percentage. To convert a ratio to a decimal, you can divide the first number in the ratio by the second number. For example, the ratio 3:2 can be expressed as a decimal as 3/2 = 1.5. To convert a ratio to a percentage, you can multiply the decimal form of the ratio by 100. For example, 1.5 * 100 = 150, so the ratio 3:2 can be expressed as a percentage as 150%.

##### How do you find the ratio of two quantities?

To find the ratio of two quantities, you must divide the first quantity by the second. For example, if you have 6 apples and 4 bananas, you can find the ratio of apples to bananas as 6/4 = 3/2. The ratio of apples to bananas is 3:2, meaning there are 3 apples for every 2 bananas.

#### Link/cite this page

If you reference any of the content on this page on your own website, please use the code below to cite this page as the original source.

<a href=»https://kidskonnect.com/math/ratio-and-proportion/»>Ratio and Proportion Worksheets: https://kidskonnect.com</a> — KidsKonnect, June 29, 2023

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**Use With Any Curriculum**

These worksheets have been specifically designed for use with any international curriculum. You can use these worksheets as-is, or edit them using Google Slides to make them more specific to your own student ability levels and curriculum standards.

## Problems on proportions in mathematics — examples with answers

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Proportions 90-60-90 have not been in trend for a long time. But what is always eternal is mathematical proportions in algebra lessons. Let’s practice and solve problems together.

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### The concept of proportion

To solve problems on the topic of proportion, let’s recall the main definition.

A proportion in mathematics is an equality between the ratios of two or more pairs of numbers or quantities.

The product of the extreme terms is equal to the product of the middle terms.
where a, b, c, d are the terms of the proportion, a, d are the extreme terms, b, c are the middle terms. |

** Derivation from the main property of the proportion: **

- The extreme term is equal to the product of the averages, which are divided by the other extreme. That is, for the proportion a/b = c/d:
- The middle term is equal to the product of the extremes, which are divided by another middle one. That is, for the proportion a/b = c/d:

To solve a proportion means to find an unknown term. The proportion property is the main assistant in the solution.

Remember!

The equality of two ratios is called proportion.

Let’s look at easy and difficult problems that can be solved using proportion. 5th, 6th, 7th, 8th grade — it doesn’t matter, it’s useful for all schoolchildren to analyze entertaining puzzles.

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### Problems on proportions with solutions and answers

Proportion properties were invented for a reason! With their help, you can find any of the terms of the proportion, if it is unknown. Let’s solve 10 proportion problems.

** Task 1. Find the unknown term of the proportion: x/2 = 3/1 **

How to solve:

In this example, the extreme term is unknown, so we multiply the middle terms and divide the result by the known extreme term:

x = (2 * 3)/1 = 6

Answer: x = 6.

** Task 2. Find the unknown term: 1/3 = 5/y **

How to solve:

y = (3 * 5)/ 1 = 15

Answer: y = 15.

** Problem 3. Solve the proportion: 30/x = 5/8 **

How to solve:

x = (30 * 8)/5 = 48

Answer: x = 48.

** Task 4. Solve: 7/5 = y/10 **

9001 4 How do we decide:

y = (7 * 10)/5 = 14

Answer: y = 14.

** Task 5. It is known that 21x = 14y. Find the ratio x — to y **

How we solve:

- First, we reduce both sides of the equality by a common factor 7: 21x/7 = 14y/7.
We get: 3x = 2y.

- Now let’s divide both sides by 3y to remove the factor 3 on the left side and get rid of y on the right side: 3x/3y = 2y/3y.
- After reducing the ratio, it turned out: x/y = 2/3.

Answer: 2 to 3.

In the following example, we will learn how to make a proportion for the task 💡

** Task 6. Out of 300 Instagram followers, 108 people liked the post. What percentage of all subscribers are those who liked the post and liked it? **

How we solve:

- Let’s take all subscribers as 100% and write down the condition of the problem briefly:
300 — 100%

108 — ?%

- Let’s make a proportion: 300/108 = 100/x.
- Find x: (108 * 100) : 300 = 36.

Answer: 36% of all subscribers liked the post.

** Task 7. Harry Potter’s girlfriend used seaweed and leeches in a ratio of 5 to 2 when brewing the Polyjuice Potion. How much seaweed do you need if there are only 450 grams of leeches? **

How we solve:

- Let’s make a proportion: 5/2 = x/450.
- Find x: (5 * 450) : 2 = 1125.

Answer: for 450 grams of leeches, you need to take 1125 grams of algae.

** Task 8. It is known that watermelon consists of 98% water. How much water is in 5 kg of watermelon? **

How we decide:

The weight of the watermelon (5 kg) is 100%. Water — 98% or x kg.

Let’s make a proportion:

5 : 100 = x : 98

x \u003d (5 * 98): 100

x \u003d 4. 9

Answer: 5 kg of watermelon contains 4.9 kg of water.

Let’s move on to more complicated examples. Let’s consider the proportion problem from the 8th grade algebra textbook.

** Task 9. Dad’s car travels from one city to another in 13 hours at a speed of 75 km/h. How long will it take him if he travels at a speed of 52 km/h? **

As we argue:

Speed and time are inversely related: the greater the speed, the less time is required.

Denote:

- v1 = 75 km/h
- v2 = 52 km/h
- t1 = 13h
- t2 = x

How we solve:

- Let’s make a proportion: v1/v2 = t2/t1.
The ratios are equal but inverted relative to each other.

- Substitute known values: 75/52 = t2/13
t2 = (75 * 13)/52 = 75/4 = 18 3/4 = 18 h 45 min

Answer: 18 hours 45 minutes.

** Task 10. 24 people promoted a Telegram channel in 5 days. In how many days will 30 people do the same job if they work with the same efficiency? **

As we argue:

1. In the completed column, put the arrow in the direction from the largest number to the smallest.

2. The more people, the less time it takes to do a certain job (channel promotion). So this is an inverse relationship.

3. So let’s point the second arrow in the opposite direction. The inverse proportion looks like this:

How we solve:

- Let 30 people can promote the channel in x days. We make a proportion:
30 : 24 = 5 : x

- To find the unknown term of the proportion, you need to divide the product of the middle terms by the known extreme term:
x = 24 * 5: 30

x = 4

- So, 30 people will promote the channel in 4 days.

Answer: 4 days.

Online preparation for the OGE in mathematics is a great way to relieve stress and consolidate knowledge before the exam.

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## Ratio and Proportion

** The basis of ** mathematical research is the ability to gain knowledge about certain quantities by comparing them with other quantities that are either * are equal to *, or * is greater than * or * is less than * than those that are the subject of the study. This is usually done using a series of * equations * and * proportions * . When we use equations, we determine the desired value by finding its * equality with some other already familiar value or quantities.
*

*
However, it often happens that we compare an unknown value with others that are not equal to but greater or less than *. Here we need a different approach to data processing. We may need to know, for example,

*how many*one value is greater than another, or

*how many times*one contains the other. To find the answer to these questions, we will find out what

*is the ratio*of two quantities. One ratio is called

*arithmetic*and the other

*geometric*. Although it is worth noting that both of these terms were not adopted by chance or just for the sake of distinction. Both arithmetic and geometric relations apply to both arithmetic and geometry.

Being a component of a vast and important subject, proportion depends on ratios, so a clear and complete understanding of these concepts is necessary.

338. ** The arithmetic ratio ** * is * ** the difference ** * between two quantities or a series of quantities * . The quantities themselves are called * members of the * ratio, that is, the terms between which there is a ratio. Thus 2 is the arithmetic ratio of 5 and 3. This is expressed by placing a minus sign between the two values, that is, 5 — 3. Of course, the term arithmetic ratio and its itemization is practically useless, since only the word 9 is replaced0297 difference to the minus sign in the expression.

339. If both terms of the arithmetic ratio * are multiplied by * or * are divided by * by the same value, then the * ratio, *, will ultimately be multiplied or divided by this value.

Thus, if we have a — b = r

Then multiply both sides by h , (Ax. 3.) ha — hb = hr

And dividing by h, (Ax. 4.) $\frac{a}{h}-\frac{b}{h}=\frac{r}{h}$

340. If the terms of an arithmetic ratio add to or subtract from the corresponding terms of another, then the ratio of the sum or difference will be equal to the sum or difference of the two ratios.

If a — b

And d — h,

are two ratios,

Then (a + d) — (b + h) = (a — b) + (d — h). Which in each case = a + d — b — h.

AND (a — d) — (b — h) = (a — b) — (d — h). Which in each case = a — d — b + h.

So the arithmetic ratio of 11 — 4 is 7

And the arithmetic ratio of 5 — 2 is 3

The ratio of the sum of the terms 16 — 6 is 10, — the sum of the ratios.

The ratio of the difference of terms 6 — 2 is 4, — the difference of ratios.

341. * The geometric ratio * is the ratio between quantities, which is expressed by

**PARTIAL**if one quantity is divided by another.

So the ratio of 8 to 4 can be written as 8/4 or 2. That is, the quotient of 8 divided by 4. In other words, it shows how many times 4 is contained in 8.

In the same way, the ratio of any quantity to another can be determined by dividing the first by the second, or, which is basically the same thing, by making the first the numerator of the fraction and the second the denominator.

So the ratio of a to b is $\frac{a}{b}$

The ratio of d + h to b + c is $\frac{d+h}{b+c}$.

342. Geometric ratio is also written by placing two points one above the other between the compared values.

So a:b is the ratio of a to b, and 12:4 is the ratio of 12 to 4. The two quantities together form a * pair * , in which the first member is called the * antecedent of * , and the last member is called the * consequent of * .

343. This dotted notation and the other, in the form of a fraction, are interchangeable as necessary, with the antecedent becoming the numerator of the fraction and the consequent the denominator.

So 10:5 is the same as $\frac{10}{5}$ and b:d is the same as $\frac{b}{d}$.

344. If any of these three values: antecedent, consequent, and relation are given any * two *, then the third one can be found.

Let a= antecedent, c= consequent, r= ratio.

By definition, $r=\frac{a}{c}$, that is, the ratio is equal to the antecedent divided by the consequent.

Multiplying by c, a = cr, that is, the antecedent is equal to the consequent times the ratio.

Divide by r, $c=\frac{a}{r}$, that is, the consequent is equal to the antecedent divided by the ratio.

Resp. 1. If two pairs have equal antecedents and consequents, then their ratios are also equal.

Resp. 2. If the ratios and antecedents of two pairs are equal, then the consequents are equal, and if the ratios and consequents are equal, then the antecedents are equal.

345. If two compared quantities are * are equal to * , then their ratio is one or the equality ratio. The ratio 3 * 6:18 is equal to one, since the quotient of any value divided by itself is equal to 1.

If the antecedent of the pair * is greater than * than the consequent, then the ratio is greater than one. Since the dividend is greater than the divisor, the quotient is greater than one. So the ratio 18:6 is 3. This is called the ratio * of the greater inequality * .

On the other hand, if the antecedent * is less than * than the consequent, then the ratio is less than one, and this is called the ratio * of the lesser inequality * . So the ratio 2:3 is less than one, because the dividend is less than the divisor.

346. ** The reciprocal ratio is the ratio of two reciprocals. **

So the inverse of 6 to 3 is ⅙ to ⅓, i.e. ⅙:⅓.

The direct relation of a to b is $\frac{a}{b}$, i.e. the antecedent divided by the consequent.

The inverse relation is $\frac{1}{a}$:$\frac{1}{b}$ or $\frac{1}{a}.\frac{b}{1}=\frac{b} {a}$.

i.e. cosequence b divided by antecedent a.

From here, the inverse ratio is expressed by * by inverting the fraction * , which displays a direct ratio, or, when notation is carried out using dots, * by inverting the order of writing * .

Thus a is related to b in the opposite way that b is related to a.

347. ** Complex ratio is the ratio of products of corresponding members with two or more simple ratios. **

So the ratio is 6:3, equal to 2

And the ratio

__12:4 is 3__

The ratio made up of them 72:12 = 6.

Here a complex relation is obtained by multiplying together two antecedents and also two consequents of simple relations.

So ratio compiled

From the ratio a:b

And ratios c:d

and ratios h:y

This is the ratio $ach:bdy=\frac{ach}{bdy}$.

A compound relation does not differ in its nature * * from any other relation. This term is used to show the origin of a relation in certain cases.

Resp. A complex ratio is equal to the product of simple ratios.

Ratio a:b equals $\frac{a}{b}$

Ratio c:d equals $\frac{c}{d}$

Ratio h:y equals $\frac{h}{y}$

And the ratio added of these three will be ach/bdy, which is the product of fractions that express simple ratios.

348. If in the sequence of relations in each previous pair the consequent is the antecedent in the next one, then * the ratio of the first antecedent and the last consequent is equal to that obtained from the intermediate relations. *

So in a number of ratios

a:b

b:c

c:d

d:h

the ratio a:h is equal to the ratio added from the ratios a:b and b:c and c:d and d:h. So the complex ratio in the last article is $\frac{abcd}{bcdh}=\frac{a}{h}$, or a:h.

In the same way, all quantities that are both antecedents and consequents * will disappear * when the product of fractions is reduced to its minor terms and in the remainder the complex relation is expressed by the first antecedent and the last consequent.

349. A special class of complex ratios is obtained by multiplying a simple ratio by * itself * or by another ratio * equal to *. These ratios are called * double * , * triple * , * quadruple * , and so on, according to the number of multiplications.

A ratio composed of * two * equal ratios, that is, * squares * of a simple ratio, is called * double * ratio.

Composed of * three * , that is, * cube * of a simple ratio, * is called triple * , and so on.

Similarly, the ratio of * square roots * of two quantities is called the ratio * of the square root of * , and the ratio of * of the cube roots of * is the ratio of * of the cube root of * , and so on.

So the simple ratio of a to b is a:b

Double ratio of a to b, equals a ^{ 2 } :b ^{ 2 }

√ Triple ratio of a to b, equal to a ^{ 3 } :b ^{ 3 }

√ √ Square root of a to b

The ratio of the cube root of a to b is ^{ 3 } √a: ^{ 3 } √b, and so on.

The terms * double * , * triple * , and so on should not be confused with * double * , * triple * , and so on.

The ratio of 6 to 2 is 6:2 = 3

If we double this ratio, that is, the ratio twice, we get 12:2 = 6

We triple this ratio, that is, this ratio three times, we get 18:2 = 9

A * double * ratio, i.e. * squared * ratio is 6 ^{ 2 } :2 ^{ 2 } = 9

AND * the triple * ratio, that is, the cube of the ratio, is 6 ^{ 3 }: 2 ^{ 3 } = 27

350. In order for quantities to be correlated with each other, they must be of the same kind, so that it can be stated with certainty whether they are equal to each other, or whether one of them is greater or less. A foot is to an inch like 12 to 1: it is 12 times larger than an inch. But one cannot, for example, say that an hour is longer or shorter than a stick, or an acre is greater or less than a degree. However, if these quantities are expressed in * numbers * , then there may be a relationship between these numbers. That is, there may be a relationship between the number of minutes in an hour and the number of steps in a mile.

351. Turning to the * nature of ratios, the next step is to take into account the way in which a change in one or two terms, which are compared with each other, will affect the ratio itself. Recall that the direct ratio is expressed as a fraction, where antecedes pairs is always numerator , and consequent — denominator . Then it will be easy to obtain from the property of fractions that changes in the ratio occur by varying the compared quantities. The ratio of the two quantities is the same as meaning fractions, each of which represents quotient : the numerator divided by the denominator. (Art. 341.) It has now been shown that multiplying the numerator of a fraction by any amount is the same as multiplying the value of by the same amount, and that dividing the numerator is the same as dividing the values of a fraction. That’s why,
*

352. * To multiply the antecedent of a pair by any value means to multiply the ratios by this value, and to divide the antecedent is to divide this ratio * .

Thus the ratio 6:2 is 3

And the ratio 24:2 is 12.

Here the antecedent and ratio in the last pair are 4 times greater than in the first.

The ratio a:b is $\frac{a}{b}$

And the relation na:b is equal to $\frac{na}{b}$.

Resp. With a known consequent, the greater than * antecedent * , the greater the * ratio * , and vice versa, the larger the ratio, the greater the antecedent.

353. * Multiplying the consequent of a pair by any value, as a result, we obtain the division of the ratio by this value, and dividing the consequent, we multiply the ratio. * By multiplying the denominator of a fraction, we divide the value, and by dividing the denominator, the value is multiplied..

So the ratio of 12:2 is 6

And the 12:4 ratio is 3.

Here is the consequent of the second pair in * is twice *, and the ratio * is twice * less than the first.

Ratio a:b equals $\frac{a}{b}$

And the ratio a:nb is $\frac{a}{nb}$.

Resp. For a given antecedent, the larger the consequent, the smaller the ratio. Conversely, the larger the ratio, the smaller the consequent.

354. It follows from the last two articles that * multiplying the antecedent * of the pair by any value will have the same effect on the ratio as * dividing the consequent * by this value, and * dividing the antecedent * will have the same effect as * multiplying the consequent * .

Therefore, the ratio 8:4 is 2

Multiplying the antecedent by 2, the 16:4 ratio is 4

Dividing the antecedent by 2, the ratio 8:2 is 4.

Resp. Any * multiplier of * or * divisor of * can be transferred from the antecedent of a pair to the consequent, or from the consequent to the antecedent, without changing the ratio.

It is worth noting that when a factor is thus transferred from one term to another, then it becomes a divisor, and the transferred divisor becomes a factor.

So ratio 3.6:9 = 2

Moving the factor 3, $6:\frac{9}{3}=2$

the same ratio.

Ratio $\frac{ma}{y}:b=\frac{ma}{by}$

Moving y $ma:by=\frac{ma}{by}$

Transferring m, a:$a:\frac{m}{by}=\frac{ma}{by}$.

355. As evident from the Articles. 352 and 353, ** if the antecedent and consequent are both multiplied or divided by the same amount, then the ratio does not change ** .

Resp. 1. The ratio of two * fractions * that have a common denominator is the same as the ratio of their * numerators * .

So the ratio a/n:b/n is the same as a:b.

Resp. 2. * The direct ratio * of two fractions that have a common numerator is equal to the inverse ratio of their * denominators * .

356. It is easy to determine the ratio of any two fractions from the article. If each term is multiplied by two denominators, then the ratio will be given by integral expressions. Thus, multiplying the members of the pair a/b:c/d by bd, we get $\frac{abd}{b}$:$\frac{bcd}{d}$, which becomes ad:bc, by reducing the total values from the numerators and denominators.

356 b. The ratio * of the greater inequality * added to another ratio, * increases * of its

Let the ratio of the greater inequality be given as 1+n:1

And any ratio like __ a:b __

A compound ratio will be (Art. 347,) a + na:b

What is greater than the ratio a:b (Art. 351 resp.)

But the ratio * of the lesser inequality * added to another ratio, * reduces * it.

Let the ratio of the lesser difference 1-n:1

Any given ratio __ a:b __

Compound ratio a — na:b

Which is less than a:b.

357. * If is added to or from the members of any pair or two other quantities that are in the same ratio are subtracted, then the sums or remainders will have the same ratio *

**.**

Let the ratio a:b

Will be the same as c:d

Then the ratio

That is, $\frac{a+c}{b+d}$ = $\frac{c}{d}$ = $\frac{a}{b}$.

Let the ratio a:b

Will be the same as c:d

Then the ratio

*of the sum of*antecedents to the sum of consequents, namely, a + c to b + d, is also the same.That is, $\frac{a+c}{b+d}$ = $\frac{c}{d}$ = $\frac{a}{b}$.

* Proof. *

1. By assumption, $\frac{a}{b}$ = $\frac{c}{d}$

2. Multiply by b and by d, ad = bc

3. Add cd to both sides, ad + cd = bc + cd

4. Divide by d, $a+c=\frac{bc+cd}{d}$

5. Divide by b + d, $\frac{a+c}{b+d}$ = $\frac{c}{d}$ = $\frac{a}{b}$.

The ratio of * difference of * antecedents to difference of consequents is also the same. 2-ab+ax-bx}{a(a+x)}$

92-ab+ax)}{a(a+x)}$.

Since the last numerator is greater than the other, then * the ratio of * is greater.

If, instead of adding the same value *, * is subtracted from two terms, then it is obvious that the effect on the ratio will be the opposite.

* Examples. *

1. Which is bigger: 11:9 ratio or 44:35 ratio?

2. Which is greater: the ratio $(a+3):\frac{a}{6}$, or the ratio $(2a+7):\frac{a}{3}$?

3. If the antecedent of a pair is 65 and the ratio is 13, what is the consequent?

4. If the consequent of a pair is 7 and the ratio is 18, what is the antecedent?

5. What does a complex ratio made up of 8:7, and 2a:5b, as well as (7x+1):(3y-2) look like?

6. What does a complex ratio composed of (x + y): b, and (x-y): (a + b), and also (a + b): h look like? Rep. (x ^{ 2 } — y ^{ 2 } ):bh.

7. If the relations (5x+7):(2x-3), and $(x+2):\left(\frac{x}{2}+3\right)$ form a complex relation, then what relation will you get: more or less inequality? Rep.