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Chapter 3: Problem 4

Write each number as a ratio of two integers.\(0.37373737 \ldots\)

Short Answer

Expert verified
The given repeating decimal is \(0.373737 \ldots\). To convert it into a ratio of two integers, we first let the number be represented by x and set up an algebraic equation: \(x = 0.373737 \ldots\). Multiply x by 100 to get \(100x = 37.3737 \ldots\). Subtract the original equation from the new one to get \(99x = 37\). Solve for x to find \(x = \frac{37}{99}\). Therefore, the repeating decimal can be represented as the ratio of two integers: \(0.373737 \ldots = \frac{37}{99}\).

Step by step solution

01

Identify the repeating part

The given number is \(0.373737 \ldots\). Notice that the repeating part of the decimal is \(37\).
02

Let the number be x and set up an equation

Let the number be represented by x: \(x = 0.373737 \ldots\) Now, to create an equation, we need to multiply x by a power of 10 that will move the repeating part to align with the same digits. In this case, we can multiply x by 100 because the repeating part has 2 digits: \(100x = 37.3737 \ldots\)
03

Subtract the original equation from the new equation

We will subtract the original equation (x) from the new equation (100x) to eliminate the repeating decimals: \(100x - x = 37.3737 \ldots - 0.373737 \ldots\) \(99x = 37\)
04

Solve for x

Now, we need to solve the equation for x: \(x = \frac{37}{99}\) So, the given repeating decimal can be represented as a ratio of two integers: \(0.373737 \ldots = \frac{37}{99}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Repeating Decimals
Repeating decimals are numbers that have a digit or a sequence of digits that repeat indefinitely after the decimal point. In our example, the number is \(0.373737\ldots\), which clearly shows the digits "37" repeating endlessly.
Understanding how these decimals behave is important for converting them into fractions.
  • To work with repeating decimals, identify the repeating sequence.
  • This sequence helps determine how many zeros will be involved when you multiply by powers of 10.
  • The goal is to align the repeating numbers effectively for subtraction in further steps.
Recognizing repeating decimals is the key to later express them as a ratio, making it easier to handle them mathematically.
Ratios
Ratios provide a way to compare two quantities by using division, and they are closely related to fractions. When a repeating decimal is expressed as a fraction, it essentially becomes a ratio of two integers. In our step-by-step solution, we converted the decimal \(0.373737\ldots\) into \(\frac{37}{99}\).
  • The ratio \(\frac{37}{99}\) signifies that for every 37 parts in one group, there are 99 equal parts in the other.
  • This representation is useful since it retains the exact value of the repeating decimal.
  • Ratios help us understand relationships in a clear and simplified form, often making them easier to work with than decimals.
Understanding how to convert a repeating decimal into a ratio broadens your ability to solve various mathematical problems effectively.
Integer Representation
Integer representation refers to expressing numbers using integers. This is particularly useful with fractions. When we look at our repeating decimal \(0.373737\ldots\), converting it to the fraction \(\frac{37}{99}\) allows us to use integers (37 and 99) for representation.
Using integer representation simplifies computations.
  • It converts a complex repeating decimal into a manageable form.
  • This transformation is crucial in algebra for solving equations.
  • It aids in eliminating the complications that arise with infinite decimal places.
Understanding how to represent numbers with integers by converting them into fractions is a foundational skill in mathematics that enhances clarity and precision.

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Most popular questions from this chapter

a. Use proof by contradiction to show that for any integer \(n\), it is impossible for \(n\) to equal both \(3 q_{1}+r_{1}\) and \(3 q_{2}+r_{2}\), where \(q_{1}, q_{2}, r_{1}\), and \(r_{2}\), are integers, \(0 \leq r_{1}<\) \(3,0 \leq r_{2}<3\), and \(r_{1} \neq r_{2}\). b. Use proof by contradiction, the quotient-remainder theorem, division into cases, and the result of part (a) to prove that for all integers \(n\), if \(n^{2}\) is divisible by 3 then \(n\) is divisible by 3 . c. Prove that \(\sqrt{3}\) is irrational.

Assume that \(m\) and \(n\) are particular integers. \(\begin{array}{ll}\text { a. Is } 6 m+8 n \text { even? } & \text { b. Is } 10 m n+7 \text { odd? }\end{array}\) c. If \(m>n>0\), is \(m^{2}-n^{2}\) composite?

If \(r\) is any rational number and \(s\) is any irrational number, then \(r / s\) is irrational.

Prove that for all positive integers \(a\) and \(b, a \mid b\) if, and only if, \(\operatorname{gcd}(a, b)=a\). (Note that to prove " \(A\) if, and only if, \(B, "\) you need to prove "if \(A\) then \(B\) " and "if \(B\) then \(A . "\) ")

The difference of any even integer minus any odd integer is odd.
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