Math / Numbers & Operations

Question10. Tomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students. His calculator showed the result as $0.9333 \ldots$ . a. Write this number as a fraction. $\frac{14}{15}$ b. How many students said that summer break should be longer?

Studdy Solution

STEP 1

1. Tomas asked 15 students in total about their opinion on whether summer break should be longer.
2. The result displayed by Tomas's calculator was $0.9333\ldots$ , which is a repeating decimal.
3. We need to convert this repeating decimal into a fraction.
4. Using the fraction found, we can determine the number of students who said yes.

STEP 2

1. Convert the repeating decimal $0.9333\ldots$ to a fraction.
2. Verify that the fraction matches $\frac{14}{15}$ .
3. Use the fraction to determine how many students said yes.

STEP 3

Let $x = 0.9333\ldots$ . This can be written as $x = 0.\overline{93}$ .

STEP 4

Multiply both sides of the equation by 10 to shift the decimal point one place to the right.
$10x = 9.3333\ldots$

STEP 5

Subtract the original equation $x = 0.9333\ldots$ from the new equation $10x = 9.3333\ldots$ .
$10x - x = 9.3333\ldots - 0.9333\ldots$

STEP 6

Simplify the left-hand side and the right-hand side of the equation.
$9x = 8.4$

STEP 7

Solve for $x$ by dividing both sides by 9.
$x = \frac{8.4}{9}$

STEP 8

Simplify the fraction $\frac{8.4}{9}$ . Since $8.4$ can be written as $\frac{84}{10}$ , we have:
$x = \frac{84}{90}$

STEP 9

Further simplify the fraction $\frac{84}{90}$ by dividing both numerator and denominator by their greatest common divisor, which is 6.
$x = \frac{84 \div 6}{90 \div 6} = \frac{14}{15}$

STEP 10

Verify that the fraction $\frac{14}{15}$ is equivalent to the repeating decimal $0.9333\ldots$ .
Since $\frac{14}{15} = 0.9333\ldots$ , this matches the calculator result.

STEP 11

Determine the number of students who said yes. Since the fraction of students who said yes is $\frac{14}{15}$ and there are 15 students in total, the number of students who said yes is:
$15 \times \frac{14}{15} = 14$
Solution: a. The fraction corresponding to $0.9333\ldots$ is $\frac{14}{15}$ . b. The number of students who said that summer break should be longer is 14.

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Studdy Solution

STEP 1

STEP 2

1. Convert the repeating decimal $0.9333\ldots$ to a fraction.
2. Verify that the fraction matches $\frac{14}{15}$ .
3. Use the fraction to determine how many students said yes.

STEP 3

Let $x = 0.9333\ldots$ . This can be written as $x = 0.\overline{93}$ .

STEP 4

Multiply both sides of the equation by 10 to shift the decimal point one place to the right.
$10x = 9.3333\ldots$

STEP 5

Subtract the original equation $x = 0.9333\ldots$ from the new equation $10x = 9.3333\ldots$ .
$10x - x = 9.3333\ldots - 0.9333\ldots$

STEP 6

Simplify the left-hand side and the right-hand side of the equation.
$9x = 8.4$

STEP 7

Solve for $x$ by dividing both sides by 9.
$x = \frac{8.4}{9}$

STEP 8

Simplify the fraction $\frac{8.4}{9}$ . Since $8.4$ can be written as $\frac{84}{10}$ , we have:
$x = \frac{84}{90}$

STEP 9

Further simplify the fraction $\frac{84}{90}$ by dividing both numerator and denominator by their greatest common divisor, which is 6.
$x = \frac{84 \div 6}{90 \div 6} = \frac{14}{15}$

STEP 10

Verify that the fraction $\frac{14}{15}$ is equivalent to the repeating decimal $0.9333\ldots$ .
Since $\frac{14}{15} = 0.9333\ldots$ , this matches the calculator result.

STEP 11

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Studdy Solution

STEP 1

STEP 2

1. Convert the repeating decimal 0.9333…0.9333\ldots0.9333… to a fraction.2. Verify that the fraction matches 1415\frac{14}{15}1514​.3. Use the fraction to determine how many students said yes.

STEP 3

Let x=0.9333…x = 0.9333\ldotsx=0.9333…. This can be written as x=0.93‾x = 0.\overline{93}x=0.93.

STEP 4

Multiply both sides of the equation by 10 to shift the decimal point one place to the right. 10x=9.3333… 10x = 9.3333\ldots 10x=9.3333…

STEP 5

Subtract the original equation x=0.9333…x = 0.9333\ldotsx=0.9333… from the new equation 10x=9.3333…10x = 9.3333\ldots10x=9.3333….10x−x=9.3333…−0.9333… 10x - x = 9.3333\ldots - 0.9333\ldots 10x−x=9.3333…−0.9333…

STEP 6

Simplify the left-hand side and the right-hand side of the equation.9x=8.4 9x = 8.4 9x=8.4

STEP 7

Solve for xxx by dividing both sides by 9.x=8.49 x = \frac{8.4}{9} x=98.4​

STEP 8

Simplify the fraction 8.49\frac{8.4}{9}98.4​. Since 8.48.48.4 can be written as 8410\frac{84}{10}1084​, we have:x=8490 x = \frac{84}{90} x=9084​

STEP 9

Further simplify the fraction 8490\frac{84}{90}9084​ by dividing both numerator and denominator by their greatest common divisor, which is 6.x=84÷690÷6=1415 x = \frac{84 \div 6}{90 \div 6} = \frac{14}{15} x=90÷684÷6​=1514​

STEP 10

Verify that the fraction 1415\frac{14}{15}1514​ is equivalent to the repeating decimal 0.9333…0.9333\ldots0.9333…. Since 1415=0.9333…\frac{14}{15} = 0.9333\ldots1514​=0.9333…, this matches the calculator result.

STEP 11

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1. Convert the repeating decimal $0.9333\ldots$ to a fraction.
2. Verify that the fraction matches $\frac{14}{15}$ .
3. Use the fraction to determine how many students said yes.

Let $x = 0.9333\ldots$ . This can be written as $x = 0.\overline{93}$ .

Multiply both sides of the equation by 10 to shift the decimal point one place to the right.
$10x = 9.3333\ldots$

Subtract the original equation $x = 0.9333\ldots$ from the new equation $10x = 9.3333\ldots$ .
$10x - x = 9.3333\ldots - 0.9333\ldots$

Simplify the left-hand side and the right-hand side of the equation.
$9x = 8.4$

Solve for $x$ by dividing both sides by 9.
$x = \frac{8.4}{9}$

Simplify the fraction $\frac{8.4}{9}$ . Since $8.4$ can be written as $\frac{84}{10}$ , we have:
$x = \frac{84}{90}$

Further simplify the fraction $\frac{84}{90}$ by dividing both numerator and denominator by their greatest common divisor, which is 6.
$x = \frac{84 \div 6}{90 \div 6} = \frac{14}{15}$

Verify that the fraction $\frac{14}{15}$ is equivalent to the repeating decimal $0.9333\ldots$ .
Since $\frac{14}{15} = 0.9333\ldots$ , this matches the calculator result.