# Reason - Meaning, Simplification, Table, Examples (2023)

Ratio, in mathematics, is a term used to compare two or more numbers. It is used to indicate how large or small one quantity is compared to another. In a ratio, two quantities are compared using division. Here the dividend is called the 'antecedent' and the divisor is called the 'consequent'. For example, in a group of 30 people, 17 prefer to walk in the morning and 13 prefer to ride a bicycle. To represent this information as a ratio, we write it as 17:13. Here, the symbol ':' is read as "is to". So the ratio of people who prefer to walk to people who prefer to bike is read as '17 is to 13'.

 1 What is the relationship? 2 Ratio Calculation 3 How to simplify reasons? 4 equivalent ratios 5 Relationship FAQ

## What is the relationship?

The ratio is defined as the comparison of two quantities of the same unitsindicating how much of one quantity is present in the other quantity. Proportions can be classified into two types. One is the part-to-part relationship and the other is the part-to-whole relationship. The part-to-part relationship indicates how two distinct entities or groups are related. For example, the ratio of boys to girls in a class is 12:15, while the ratio of part to whole denotes the relationship between a specific group and a whole. For example, out of 10 people, 5 like to read books. Therefore, the ratio of the part to the whole is 5:10, which means that every 5 people out of 10 people like to read books.

### account formula

we use thesystem formulawhen comparing the relationship between two numbers or quantities. The general way to represent a relationship between two quantities, say 'a' and 'b' isa: b,which is read as'a is a b'. The fraction form that represents this ratio is a/b. To further simplify a ratio, we follow the same procedure that we use to simplify a fraction. a:b = a/b. Let's understand this with an example.

Example:In a class of 50 students, 23 are girls and the rest are boys. Find the ratio between the number of boys and the number of girls.

Total number of students = 50; Number of girls = 23.

(Video) Truth Table Tutorial - Discrete Mathematics Logic

Total number of boys = Total number of students - Total number of girls
= 50 - 23
= 27

Therefore, the desired ratio is, (Number of boys: Number of girls), which is27:23.

## Ratio Calculation

To calculate the ratio of two quantities, we can use the following steps. Let's understand this with an example. For example, if 15 cups of flour and 20 cups of sugar are needed to make fluffy pancakes, let's calculate the ratio of flour and sugar used in the recipe.

• Paso 1:Find the quantities of both scenarios for which we are determining the ratio. In this case, they are 15 and 20.
• Paso 2:Write it as a fraction a/b. So, we write it as 15/20.
• Paso 3:simplify the fractionfurther, if possible. The simplified fraction will give the final proportion. Here, 15/20 can be simplified to 3/4.
• Stage 4:Therefore, the ratio of flour to sugar can be expressed as 3:4.

## How to simplify reasons?

A ratio expresses how much of one quantity is required compared to another quantity. The two terms of the ratio can be simplified and expressed in their minimal form. Ratios when expressed in their lowest terms are easy to understand and can be simplified in the same way that we simplify fractions. To simplify a ratio, we use the following steps. Let's understand this with an example. For example, let's simplify the ratio 18:10.

• Paso 1:Write the given ratio a:b as a fraction a/b. Writing the ratio as a fraction, we get 18/10.
• Paso 2:find thegreatest common divisorof 'a' and 'b'. In this case, the LCD of 10 and 18 is 2.
• Paso 3:Divide the numerator and denominator of the fraction with the LCD to get the simplified fraction. Here, dividing the numerator and denominator by 2, we get (18 ÷ 2)/(10 ÷ 2) = 9/5.
• Stage 4:Represent this fraction as a ratio to get the result. Therefore, the simplified ratio is 9:5.

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Proportion Tips and Tricks:

• In case the numbers 'a' and 'b' are equal in the ratio a:b, then a:b = 1.
• If a > b in the ratio a : b, then a : b > 1.
• If a < b in the ratio a : b, then a : b < 1.
• You must make sure that the units of the two quantities are similar before you compare them.

## equivalent ratios

Equivalent ratios are similar to equivalent fractions. If the antecedent (the first term) and the consequent (the second term) of a given ratio are multiplied or divided by the same nonzero number, it gives aequivalent relationship. For example, when the antecedent and consequent of the ratio 1:3 are multiplied by 3, we get (1 × 3): (3 × 3) or 3: 9. Here, 1:3 and 3:9 are equivalent ratios. Similarly, when both terms of the ratio 20:10 are divided by 10, it gives 2:1. Here, 20:10 and 2:1 are equal ratios. An infinite number of equivalent ratios of any given ratio can be found by multiplying the antecedent and consequent by apositive integer.

### ratio table

Aratio tableis a list containing the equivalent quotients of any given quotient in a structured way. The ratio table below gives the relationship between the ratio 1:4 and four of its equivalent ratios. Equivalent ratios are related to each other by themultiplicationof a number Equivalent ratios are obtained by multiplying or dividing the two terms of a ratio by the same number. In the example shown in the figure, let's take the ratio 1:4 and find four equivalent ratios by multiplying both terms of the ratio by 2, 3, 6, and 9. As a result, we get 2:8, 3:12, 6 :24 and 9:36. ☛ Related topics

• percentage ratio
• Rate Definition
• Return Rate Calculator

## Relationship FAQ

### What is reason in mathematics?

A ratio can be defined as the relationship or comparison between two numbers of the same unit to check how much one number is greater than the other. For example, if the number of points earned on a test is 7 out of 10, the ratio of points earned to the total number of points is written as 7:10.

### What are the ways to write a ratio?

A ratio can be written by separating the two quantities using a colon (:) or it can be written in fractional form. For example, if there are 4 apples and 8 melons, then the ratio of apples to melons can be written as 4:8 or 4/8, which can be further simplified as 1:2.

### How to calculate the ratio between two numbers?

To calculate the ratio of two quantities, we can use the following steps. Let's understand this with an example. For example, if 14 cups of butter and 28 cups of sugar are needed to make a frosting, what is the ratio of butter to sugar?

• Paso 1:Note the amounts of both ingredients for which we are determining the ratio. In this case, they are 14 and 28.
• Paso 2:Write it as a fraction a/b. So, we write it as 14/28.
• Paso 3:Simplify the fraction even more, if possible. The simplified fraction will give the final proportion. Here, 14/28 can be simplified to 1/2.
• Stage 4:Therefore, the ratio of butter to sugar can be expressed as 1:2.

### How to find equivalent ratios?

Two ratios are said to be equivalent if they represent the same value when simplified. This concept is similar toequivalent fractions. For example, when the ratio 1: 4 is multiplied by 2, it means multiplying both terms of the ratio by 2. So, we get, (1 × 2)/ (4 × 2) = 2/8 or 2: 8 Here, 1 :4 and 2:8 are equivalent ratios. Similarly, the ratio 30:10, when divided by 10, results in a ratio of 3:1. Here, 30:10 and 3:1 are equal ratios. So, the equivalent ratios can be found using multiplication ordivisionoperation depending on the numbers.

### What is a ratio table?

A ratio table shows a list of equivalent ratios that are obtained by multiplying or dividing both quantities by the same value. For example, if the ratio table starts with the ratio 1:3, successive rows will have 2:6, 3:9, 4:12, etc. When these ratios are simplified, they represent the same value, i.e. 1:3.

### What is the golden ratio?

Agolden ratiois a distinct number whose value is approximately equal to 1.618. The symbol for this is a Greek letter 'phi' represented as ϕ. It is a special attribute and is used in art, geometry, and architecture because the golden ratio is believed to create the most pleasing and beautiful shape. It is also known as the divine proportion that exists between two quantities and the relationship to calculate the golden ratio is represented as ϕ = a/b = (a + b)/a = 1.61803398875... where a and b are the dimensions of two quantities and a is the older of the two.

### Why are proportions important?

Ratios are important because they allow us to express quantities in such a way that they are easier to interpret. It is a tool used to compare the size of two or more quantities with each other. For example, if there are 30 girls and 20 boys in a class. We can represent the number of girls to the number of boys with the help of the ratio which is 3:2 in this case.

(Video) Conditional Statements: if p then q

### What is the ratio formula?

The ratio formula is used to compare the relationship between twonumbersor amounts. The general way to represent a relationship between two quantities, say 'a' and 'b' isa: b,which is read as'a is a b'.

### What is ratio and proportion?

The ratio is the relationship or comparison between two quantities of the same unit to see how much one number is greater than the other. It is written as a/b or a:b where b is not equal to zero. TOproportionis an equality of two ratios. Proportions are used to write equivalent ratios that help solve for unknown quantities. For example, a proportion is expressed as, a:b = c:d

### How to compare proportions?

There are several methods tocompare proportions. For example, let's compare 1:2 and 2:3 using the LCM method.

• Step 1: Write the ratios as a fraction. Here, it means 1/2 and 2/3.
• Paso 2:reduce fractionsseparately. Here, both the fractions 1/2 and 2/3 are already in their reduced form.
• Step 3: Now, compare 1/2 and 2/3 by finding themcm (least common multiple)of the denominators. The LCM of 2 and 3 is 6.
• Step 4: Match the denominators by multiplying the numerator and denominator of the first fraction by 3, that is, (1 × 3)/(2 × 3) = 3/6. Then, multiply the numerator and denominator of the second fraction by 2, that is, (2 × 2)/(3 × 2) = 4/6.
• Step 5 – Now 3/6 and 4/6 can be easily compared. This shows that 4/6 is greater than 3/6. Therefore, 2:3 > 1:2.

### How to convert ratios to fractions?

The ratios can be written as fractions in a very simple way. The antecedent is written as thenumeratorand the consequent is written as the denominator. For example, if we take the ratio 3:5. Here, 3 is the antecedent and 5 is the consequent. So we can write it as 3/5.

### How to convert fractions to ratios?

fractionscan be written in the form of ratios after simplification. This means that we first reduce the given fraction to its lowest terms and then write the numerator as the antecedent and thedenominatoras the consequent. For example, the fraction 16/48 will first be reduced to 1/3 and can then be expressed as a ratio as 1:3.

### How to convert ratios to decimals?

The proportions can easily be converted todecimalswriting the ratio as a fraction, and then converting the fraction to a decimal by dividing the numerator by the denominator. For example, 3:7 can be written as 3/7. Now, 3/7 = 0.428.

### How to convert ratios to percentages?

Ratios can be converted to percentages by following the steps below. For example, let's transform 5:6 into the form of apercentage.

• Step 1: Write the ratio as a fraction. Here, 5: 6 can be written as 5/6.
• Step 2: Multiply this fraction by 100 and add the percent symbol. In this case, 5/6 × 100 = 83.33%.

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## FAQs

### Is p → q ∧ q → p logically equivalent to p → q ∨ q ↔ p? ›

Look at the following two compound propositions: p → q and q ∨ ¬p. (p → q) and (q ∨ ¬p) are logically equivalent. So (p → q) ↔ (q ∨ ¬p) is a tautology. Thus: (p → q)≡ (q ∨ ¬p).

Is the statement p → q ≡ p ∧ q logically equivalent? ›

Theorem 2.6. For statements P and Q, The conditional statement P→Q is logically equivalent to ⌝P∨Q. The statement ⌝(P→Q) is logically equivalent to P∧⌝Q.

What does P → Q mean? ›

The implication p → q (read: p implies q, or if p then q) is the state- ment which asserts that if p is true, then q is also true. We agree that p → q is true when p is false. The statement p is called the hypothesis of the implication, and the statement q is called the conclusion of the implication.

What does P ∧ Q mean? ›

● Logical AND: p ∧ q

Read “p and q.” ● p ∧ q is true if both p and q are true. ● Also called logical conjunction.

Is the statement p ∧ q ∨ r equivalent to p ∧ q ∨ r explain? ›

Since columns corresponding to p∨(q∧r) and (p∨q)∧(p∨r) match, the propositions are logically equivalent. This particular equivalence is known as the Distributive Law.

What is logically equivalent to ~( P → Q? ›

P → Q is logically equivalent to ¬ P ∨ Q . Example: “If a number is a multiple of 4, then it is even” is equivalent to, “a number is not a multiple of 4 or (else) it is even.”

What kind of proposition is p → q ∨ q → p? ›

The biconditional statement is equivalent to (p → q) ∧ (q → p). In other words, for p ↔ q to be true we must have both p and q true or both false.

Is P → Q → [( P → Q → Q a tautology Why or why not? ›

Which one of the following is not equivalent to p ↔ q? ›

Option (c) is p′q + pq′ = p ⨁ p which is not equaivalent to p↔q.

What does ∧ and ∨ mean in math? ›

The conjunction of the statements P and Q is the statement “P and Q” and its denoted by P∧Q. The statement P∧Q is true only when both P and Q are true. The disjunction of the statements P and Q is the statement “P or Q” and its denoted by P∨Q. The statement P∨Q is true only when at least one of P or Q is true.

### What is the truth value of P ∨ q? ›

Since p∨q and p∧q both are T, from the table, the truth values of both p and q are T.

Is P → q ∨ q → pa tautology? ›

The given statement is a tautology as the truth table has all the values as true in the output which is the property of tautology.

What is the conditional statement P → q? ›

The conditional statement P→Q means that Q is true whenever P is true. It says nothing about the truth value of Q when P is false. Using this as a guide, we define the conditional statement P→Q to be false only when P is true and Q is false, that is, only when the hypothesis is true and the conclusion is false.

What is the conditional law for P → q? ›

The conditional of q by p is "If p then q" or "p implies q" and is denoted by p q. It is false when p is true and q is false; otherwise it is true. Contrapositive: The contrapositive of a conditional statement of the form "If p then q" is "If ~q then ~p".

What does ∨ mean in math? ›

\parallel. logical (inclusive) disjunction. or. propositional logic, Boolean algebra. The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.

What is inverse P and Q? ›

The inverse of a conditional statement is when both the hypothesis and conclusion are negated; the “If” part or p is negated and the “then” part or q is negated. In Geometry the conditional statement is referred to as p → q. The Inverse is referred to as ~p → ~q where ~ stands for NOT or negating the statement.

Is P and Q implies P is a tautology? ›

Clearly from the truth table, we can conclude that the truth values of p∨(p→q) and (p∧q)→q are always true. Hence, they are tautology.

What is PQ → R logically equivalent to? ›

Solution. (p ∧ q) → r is logically equivalent to p → (q → r).

What is the negation of P → Q? ›

The negation of compound statements works as follows: The negation of “P and Q” is “not-P or not-Q”. The negation of “P or Q” is “not-P and not-Q”.

Is P → QVP a tautology a contingency or a contradiction? ›

As seen above, 'P v Q' is a contingent statement – there are instances where it is true (row 1, 2 and 3), and an instance where it is false (row 4).

### What is an example of if p then q? ›

"If p, then not q" is equivalent to "No p are q." Example: "If something is a poodle, then it is a dog" is a round-about way of saying "All poodles are dogs."

How do you tell if a statement is a tautology? ›

One way to determine if a statement is a tautology is to make its truth table and see if it (the statement) is always true. Similarly, you can determine if a statement is a contradiction by making its truth table and seeing if it is always false.

Is P → true a tautology? ›

Hence, ``[Not(P)] or P'' is always true.

So, "if P, then P" is also always true and hence a tautology.

What is the relation between Q and P? ›

The relation between p and q is represented by the equation p = q − 4.

Is the inverse of p → q the conditional statement q →? ›

Converse, Inverse, Contrapositive
StatementIf p , then q .
ConverseIf q , then p .
InverseIf not p , then not q .
ContrapositiveIf not q , then not p .

What is the Contrapositive of P ∨ Q → R? ›

∴ Contrapositive of (p∨q)⇒r is ∼r⇒∼(p∨q) i.e. ∼r⇒(∼p∧∼q).

What is the converse of p → q → r? ›

The converse of p→(q→r) is. Simply (a^(p)/a^(q))^(p+q-r).

Is P → Q and Q → R are true conditionals then P → R is also true? ›

The law of syllogism tells us that if p → q and q → r then p → r is also true.

Can P imply not P? ›

Negation: if p is a statement variable, the negation of p is "not p", denoted by ~p. If p is true, then ~p is false.

Is P the consequent in the conditional statement if P then Q? ›

A conditional is a logical compound statement in which a statement p, called the antecedent, implies a statement q, called the consequent. A conditional is written as p→q and is translated as "if p, then q".

### What are these symbols called in English * {} []? ›

Curly brackets {}

Curly brackets, also known as braces, are rarely used punctuation marks that are used to group a set.

What is ⊕ in math? ›

⊕︀ (logic) exclusive or. (logic) intensional disjunction, as in some relevant logics. (mathematics) direct sum. (mathematics) An operator indicating special-defined operation that is similar to addition.

What does ∃ mean in math? ›

The symbol ∀ means “for all” or “for any”. The symbol ∃ means “there exists”.

Is P ∧ Q → P → Q a tautology? ›

Clearly from the truth table, we can conclude that the truth values of p∨(p→q) and (p∧q)→q are always true. Hence, they are tautology.

Is P → Q ∨ Q → pa tautology? ›

The given statement is a tautology as the truth table has all the values as true in the output which is the property of tautology.

Why is P implies Q the same as not P or Q? ›

The statement “p implies q” means that if p is true, then q must also be true. The statement “p implies q” is also written “if p then q” or sometimes “q if p.” Statement p is called the premise of the implication and q is called the conclusion.

Which is logically equivalent to p∧q → r? ›

Solution. (p ∧ q) → r is logically equivalent to p → (q → r).

What is the inverse of p → Q? ›

Suppose a conditional statement of the form "If p then q" is given. The inverse is "If ~p then ~q." Symbolically, the inverse of p q is ~p ~q.

Is p → QVP a tautology a contingency or a contradiction? ›

As seen above, 'P v Q' is a contingent statement – there are instances where it is true (row 1, 2 and 3), and an instance where it is false (row 4).

What are the truth values for ~( p ∨ q? ›

We know that p∨q is only true when at least one of p or q are true. We know that p→q is only false when both p is true and q is false.

### What is an example of a tautology in a truth table? ›

For example for any two given statements such as x and y, (x ⇒ y) ∨ (y ⇒ x) is a tautology. The simple examples of tautology are; Either Mohan will go home or Mohan will not go home. He is healthy or he is not healthy.

What is the statement P and P implies Q implies Q? ›

p → q (p implies q) (if p then q) is the proposition that is false when p is true and q is false and true otherwise.

What is the negation of the statement p → q ∨ r? ›

So, ∼(p→(q ∨ r))≡ p ∧(∼q∧∼r)

What is the negation of P → PV Q? ›

Solution. The negation of p → (~p ∨ q) is P ∧ ∼q.

What is the equivalent of if not p then not q? ›

"If p, then not q" is equivalent to "No p are q." Example: "If something is a poodle, then it is a dog" is a round-about way of saying "All poodles are dogs."

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