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Chapter 7: Problem 84

Write as a percent. Write the remainder in fractional form. $$\frac{3}{7}$$

Short Answer

Expert verified
\(\frac{3}{7}\) is 42.8% with a remainder of \(\frac{6}{7}\).

Step by step solution

01

Understand the Problem

We need to convert the fraction \(\frac{3}{7}\) to a decimal and then express it as a percentage. Additionally, any remainder must be written as a fraction.
02

Convert Fraction to Decimal

Divide the numerator (3) by the denominator (7) to convert the fraction into a decimal. Perform the division: \(3 \div 7 = 0.428571428571...\). Since it's a repeating decimal, for initial calculations, use the non-repeating part: 0.428.
03

Convert Decimal to Percentage

Multiply the decimal by 100 to convert it into a percentage. \(0.428 \times 100 = 42.8\%\). This represents the percentage equivalent of \(\frac{3}{7}\).
04

Identify the Remainder and Express as Fraction

In the division \(3 \div 7\), we identified 0.428 as the main decimal part. Calculate the remainder when 3 is divided by 7 using long division. The remainder from this division is 6, so the remaining part of \(\frac{3}{7}\) is \(\frac{6}{7}\).

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Key Concepts

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

Decimal Conversion
Converting a fraction into a decimal involves dividing the numerator by the denominator. For the fraction \( \frac{3}{7} \), this means dividing 3 by 7. When performing this division, we encounter a decimal that "never ends" and starts to repeat. This is called a repeating decimal.
  • Start by placing the numerator inside a long division bracket and the denominator outside.
  • If necessary, add a decimal point and zeros to the dividend, so the division can be carried out to as many places as needed.
  • Divide as you would with integers, but note where the decimal repeats.
For \( \frac{3}{7} \), dividing gives a repeating decimal: approximately 0.428571, which continuously repeats. When written with this repeating sequence, it is important to let the reader know that the decimal continues indefinitely.
Repeating Decimals
Repeating decimals occur when a fraction does not divide evenly, and the digits after the decimal point begin to repeat in a set pattern. In our example with \(\frac{3}{7}\), the repeating decimal is 0.428571. This pattern keeps repeating indefinitely, marked as 0.428571 by placing a bar over the sequence.
  • Some fractions will convert to terminating decimals, like 0.5, but in cases such as \(\frac{3}{7}\), this does not happen.
  • Understanding repeating patterns helps ensure accuracy in calculations.
  • Knowing when a decimal repeats is key in rounding or when using approximations.
For initial conversions, it's often practical to use the first few digits, especially if an exact value isn't necessary. For \(\frac{3}{7}\), using 0.428 for calculations as a decimal is sufficient for many basic applications.
Long Division
Long division is a process of dividing larger numbers that is foundational to converting fractions to decimals. When dividing 3 by 7, the long division method helps identify the repeating decimal and any remainder.
  • Set up the division by writing the dividend (3) under the division bar and the divisor (7) outside to the left.
  • Add a period after the 3 and append enough zeros to carry out the division.
  • Proceed with the division as normal: divide, multiply, subtract, and bring down the next digit.
During this operation, you reach a point where the partial remainders start repeating. For 3 divided by 7, you will quickly find the remainder cycles through the same numbers, creating the pattern 428571. The important part is to recognize when the cycle begins repeating, allowing you to say the decimal part is 0.428571428571... with the repeating sequence acknowledged.

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Most popular questions from this chapter

Divide. $$0.875 \div 0.4375$$

Find the difference of \(\frac{8}{15}\) and \(\frac{8}{35}\).

The problems below will allow you to review subtraction of fractions and mixed numbers. $$\frac{9}{12}-\frac{1}{5}$$

Multiply. $$15 \div 5$$

Use a calculator to write each fraction as a decimal, and then change the decimal to a percent. Round all answers to the nearest tenth of a percent. $$\frac{18}{83}$$
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