Fill In The Blanks To Make The Equation True:${ \frac{5}{6} = \frac{x}{12} = \frac{\square \cdot \square}{\square} = \frac{\square}{\square} = \square }$

Introduction

Mathematics is a fascinating subject that involves solving equations and inequalities to find the unknown values. In this article, we will explore a mathematical equation that involves fractions and variables. The equation is given as:

$$\frac{5}{6} = \frac{x}{12} = \frac{\square \cdot \square}{\square} = \frac{\square}{\square} = \square$$

Our goal is to fill in the blanks to make the equation true. We will use algebraic manipulations and mathematical properties to solve for the unknown values.

Understanding the Equation

The given equation involves four fractions, each of which is equal to the others. This means that we can set up a series of equalities to represent the relationships between the fractions.

Let's start by analyzing the first two fractions:

$$\frac{5}{6} = \frac{x}{12}$$

We can cross-multiply to get:

$$5 \cdot 12 = 6 \cdot x$$

Simplifying the equation, we get:

$$60 = 6x$$

Dividing both sides by 6, we get:

$$x = 10$$

Filling in the Blanks

Now that we have found the value of x, we can fill in the blanks in the original equation.

$$\frac{5}{6} = \frac{10}{12} = \frac{\square \cdot \square}{\square} = \frac{\square}{\square} = \square$$

We can see that the first two fractions are already equal to the value of x. Now, let's focus on the third fraction:

$$\frac{\square \cdot \square}{\square}$$

We know that the product of two numbers is equal to the product of the first number and the second number. Therefore, we can write:

$$\frac{a \cdot b}{c} = \frac{a}{c} \cdot \frac{b}{1}$$

Applying this property to the third fraction, we get:

$$\frac{\square \cdot \square}{\square} = \frac{a}{c} \cdot \frac{b}{1}$$

We can see that the first fraction is equal to the value of x, which is 10. Therefore, we can write:

$$\frac{10}{12} = \frac{a}{c} \cdot \frac{b}{1}$$

Simplifying the equation, we get:

$$\frac{5}{6} = \frac{a}{c}$$

Finding the Value of a and c

We know that the first fraction is equal to the value of x, which is 10. Therefore, we can write:

$$\frac{5}{6} = \frac{a}{c}$$

We can cross-multiply to get:

$$5 \cdot c = 6 \cdot a$$

Simplifying the equation, we get:

$$5c = 6a$$

We can see that the left-hand side of the equation is a multiple of 5, while the right-hand side is a multiple of 6. This means that the values of a and c must be multiples of 5 and 6, respectively.

Let's assume that a = 5k and c = 6k, where k is a constant. Substituting these values into the equation, we get:

$$5 \cdot 6k = 6 \cdot 5k$$

Simplifying the equation, we get:

$$30k = 30k$$

This means that the values of a and c are indeed multiples of 5 and 6, respectively.

Finding the Value of b

We know that the first fraction is equal to the value of x, which is 10. Therefore, we can write:

$$\frac{5}{6} = \frac{a}{c} \cdot \frac{b}{1}$$

We can see that the first fraction is equal to the value of x, which is 10. Therefore, we can write:

$$\frac{5}{6} = \frac{a}{c} \cdot \frac{b}{1}$$

Simplifying the equation, we get:

$$\frac{5}{6} = \frac{5k}{6k} \cdot \frac{b}{1}$$

We can cancel out the common factors to get:

$$\frac{5}{6} = \frac{b}{1}$$

This means that the value of b is equal to 5.

Conclusion

In this article, we filled in the blanks to make the equation true. We used algebraic manipulations and mathematical properties to solve for the unknown values. We found that the value of x is 10, and the values of a, b, and c are 5, 5, and 6, respectively.

The final answer is:

$$\frac{5}{6} = \frac{10}{12} = \frac{5 \cdot 5}{6} = \frac{5}{6} = 5$$

References

• [1] Algebraic Manipulations, Math Open Reference
• [2] Mathematical Properties, Khan Academy
• [3] Fractions, Math Is Fun
  

Introduction

In our previous article, we explored a mathematical equation that involved fractions and variables. We filled in the blanks to make the equation true and found the values of x, a, b, and c. In this article, we will answer some frequently asked questions (FAQs) related to the equation.

Q: What is the value of x in the equation?

A: The value of x is 10.

Q: How did you find the value of x?

A: We cross-multiplied the first two fractions to get:

$$5 \cdot 12 = 6 \cdot x$$

Simplifying the equation, we got:

$$60 = 6x$$

Dividing both sides by 6, we got:

$$x = 10$$

Q: What is the value of a in the equation?

A: The value of a is 5.

Q: How did you find the value of a?

A: We assumed that a = 5k and c = 6k, where k is a constant. Substituting these values into the equation, we got:

$$5 \cdot 6k = 6 \cdot 5k$$

Simplifying the equation, we got:

$$30k = 30k$$

This means that the values of a and c are indeed multiples of 5 and 6, respectively.

Q: What is the value of b in the equation?

A: The value of b is 5.

Q: How did you find the value of b?

A: We simplified the equation:

$$\frac{5}{6} = \frac{a}{c} \cdot \frac{b}{1}$$

We canceled out the common factors to get:

$$\frac{5}{6} = \frac{b}{1}$$

This means that the value of b is equal to 5.

Q: What is the value of c in the equation?

A: The value of c is 6.

Q: How did you find the value of c?

A: We assumed that a = 5k and c = 6k, where k is a constant. Substituting these values into the equation, we got:

$$5 \cdot 6k = 6 \cdot 5k$$

Simplifying the equation, we got:

$$30k = 30k$$

This means that the values of a and c are indeed multiples of 5 and 6, respectively.

Q: Can you explain the concept of cross-multiplication?

A: Cross-multiplication is a technique used to solve equations involving fractions. It involves multiplying the numerator of one fraction by the denominator of another fraction, and vice versa.

For example, in the equation:

$$\frac{5}{6} = \frac{x}{12}$$

We can cross-multiply to get:

$$5 \cdot 12 = 6 \cdot x$$

Simplifying the equation, we get:

$$60 = 6x$$

Dividing both sides by 6, we get:

$$x = 10$$

Q: Can you explain the concept of algebraic manipulations?

A: Algebraic manipulations are techniques used to solve equations involving variables. They involve using mathematical properties and operations to simplify and solve the equation.

For example, in the equation:

$$\frac{5}{6} = \frac{a}{c} \cdot \frac{b}{1}$$

We can simplify the equation by canceling out the common factors:

$$\frac{5}{6} = \frac{b}{1}$$

This means that the value of b is equal to 5.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the equation. We explained the concepts of cross-multiplication and algebraic manipulations, and provided examples of how to use these techniques to solve the equation.

The final answer is:

$$\frac{5}{6} = \frac{10}{12} = \frac{5 \cdot 5}{6} = \frac{5}{6} = 5$$

References

• [1] Algebraic Manipulations, Math Open Reference
• [2] Mathematical Properties, Khan Academy
• [3] Fractions, Math Is Fun