Which Of The Following Is Exactly Equal To 1?A. The Reciprocal Of 3 15 \frac{3}{15} 15 3 ​ B. The Opposite Of 3 15 \frac{3}{15} 15 3 ​ Times 3 15 \frac{3}{15} 15 3 ​ C. 15 3 \frac{15}{3} 3 15 ​ Divided By 15 3 \frac{15}{3} 3 15 ​ D. None Of These

Introduction

In mathematics, the concept of equality is crucial in understanding various mathematical operations and relationships. When it comes to fractions, equality can be determined by comparing the values of different fractions. In this article, we will explore which of the given options is exactly equal to 1.

Understanding the Options

A. The Reciprocal of $\frac{3}{15}$

The reciprocal of a fraction is obtained by swapping its numerator and denominator. In this case, the reciprocal of $\frac{3}{15}$ is $\frac{15}{3}$. To determine if this is equal to 1, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3.

$\frac{15}{3} = \frac{15 \div 3}{3 \div 3} = \frac{5}{1}$

As we can see, the reciprocal of $\frac{3}{15}$ is indeed equal to 1.

B. The Opposite of $\frac{3}{15}$ Times $\frac{3}{15}$

The opposite of a fraction is obtained by changing its sign. In this case, the opposite of $\frac{3}{15}$ is $-\frac{3}{15}$. When we multiply this by $\frac{3}{15}$, we get:

$-\frac{3}{15} \times \frac{3}{15} = -\frac{9}{225}$

To simplify this fraction, we can divide both the numerator and denominator by their GCD, which is 9.

$-\frac{9}{225} = -\frac{9 \div 9}{225 \div 9} = -\frac{1}{25}$

As we can see, the opposite of $\frac{3}{15}$ times $\frac{3}{15}$ is not equal to 1.

C. $\frac{15}{3}$ Divided by $\frac{15}{3}$

When we divide a fraction by another fraction, we can multiply the first fraction by the reciprocal of the second fraction. In this case, we can rewrite the division as a multiplication:

$\frac{15}{3} \div \frac{15}{3} = \frac{15}{3} \times \frac{3}{15}$

When we multiply these fractions, we get:

$\frac{15}{3} \times \frac{3}{15} = \frac{45}{45}$

To simplify this fraction, we can divide both the numerator and denominator by their GCD, which is 45.

$\frac{45}{45} = \frac{45 \div 45}{45 \div 45} = \frac{1}{1}$

As we can see, $\frac{15}{3}$ divided by $\frac{15}{3}$ is indeed equal to 1.

D. None of These

Based on our analysis of the previous options, we can see that option A and option C are both equal to 1. Therefore, option D is incorrect.

Conclusion

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Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of $\frac{3}{15}$ is $\frac{15}{3}$.

Q: How do I simplify a fraction?

A: To simplify a fraction, you can divide both the numerator and denominator by their greatest common divisor (GCD). For example, to simplify $\frac{15}{3}$, you can divide both the numerator and denominator by 3:

$\frac{15}{3} = \frac{15 \div 3}{3 \div 3} = \frac{5}{1}$

Q: What is the opposite of a fraction?

A: The opposite of a fraction is obtained by changing its sign. For example, the opposite of $\frac{3}{15}$ is $-\frac{3}{15}$.

Q: How do I multiply fractions?

A: To multiply fractions, you can multiply the numerators and denominators separately. For example, to multiply $\frac{3}{15}$ and $\frac{3}{15}$, you can multiply the numerators and denominators separately:

$\frac{3}{15} \times \frac{3}{15} = \frac{3 \times 3}{15 \times 15} = \frac{9}{225}$

Q: How do I divide fractions?

A: To divide fractions, you can multiply the first fraction by the reciprocal of the second fraction. For example, to divide $\frac{15}{3}$ by $\frac{15}{3}$, you can multiply the first fraction by the reciprocal of the second fraction:

$\frac{15}{3} \div \frac{15}{3} = \frac{15}{3} \times \frac{3}{15}$

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, $\frac{3}{15}$ is a fraction. A decimal is a way of expressing a number as a sum of powers of 10. For example, 0.2 is a decimal.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you can divide the numerator by the denominator. For example, to convert $\frac{3}{15}$ to a decimal, you can divide the numerator by the denominator:

$\frac{3}{15} = \frac{3 \div 15}{15 \div 15} = 0.2$

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, you can express the decimal as a sum of powers of 10 and then simplify the resulting fraction. For example, to convert 0.2 to a fraction, you can express the decimal as a sum of powers of 10:

$0.2 = \frac{2}{10} = \frac{1}{5}$

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 15 and 3 is 3.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the Euclidean algorithm. The Euclidean algorithm is a method for finding the GCD of two numbers by repeatedly dividing the larger number by the smaller number and taking the remainder.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 15 and 3 is 15.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that is a multiple of both numbers.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a ratio of two integers. For example, $\frac{3}{15}$ is a rational number. An irrational number is a number that cannot be expressed as a ratio of two integers. For example, $\sqrt{2}$ is an irrational number.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you can try to express the number as a ratio of two integers. If you can express the number as a ratio of two integers, then it is a rational number. If you cannot express the number as a ratio of two integers, then it is an irrational number.