$2$ . What number must be subtracted from $67345643$ to get $54903891$ ?

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Grade 8

Reciprocals and multiplicative inverses

Full solution

2

. What number must be subtracted from

67345643

to get

54903891

Identify Meaning of Bar Notation: Identify the meaning of the bar notation. The bar over the number $4903891$ indicates that the digits under the bar are repeating. So, $5\overline{4903891}$ represents the number $5.49038913901389139013891\ldots$ and so on, with the sequence $4903891$ repeating indefinitely.
Convert to Fraction: Recognize that to subtract and find the original number, we need to work with a finite representation of the repeating decimal. Let's convert $5\overline{4903891}$ to a fraction. To do this, let $x = 5.490389139013891\ldots$ , then multiply $x$ by $10^7$ (which is $10000000$ ) to shift the decimal point $7$ places to the right, getting $10^7x = 54903891.4903891\ldots$
Subtract Original $x$ : Subtract the original $x$ from the $10^7x$ to get rid of the repeating part. This gives us $10^7x - x = 54903891.4903891... - 5.4903891... = 54903886$ .
Simplify Left Side: Now, simplify the left side of the equation: $10^7x - x = (10^7 - 1)x = 9999999x$ .
Solve for x: Solve for x by dividing both sides of the equation by $9999999$ : x = rac{54903886}{9999999}.
Perform Division: Perform the division to find the value of $x$ : $x = \frac{54903886}{9999999} = 5 + \frac{4903891}{9999999}$ .
Subtract Fraction from Whole Number: Now that we have the repeating decimal as a fraction, we can subtract it from $67345643$ . Let's represent the fraction as $F$ for simplicity: $F = \frac{4903891}{9999999}$ . The subtraction we want to perform is $67345643 - F$ .
Substitute Fraction: Substitute the fraction back into the subtraction: $67345643 - \frac{4903891}{9999999}.$
Write Whole Number as Fraction: To subtract the fraction from the whole number, we can write the whole number as a fraction with the same denominator: $\frac{67345643 \times 9999999}{9999999} - \frac{4903891}{9999999}$ .
Perform Subtraction of Numerators: Perform the subtraction of the numerators while keeping the common denominator: $[(67345643 \times 9999999) - 4903891] / 9999999$ .
Calculate Numerator: Calculate the numerator: $(67345643 \times 9999999) - 4903891 = 673456429999999 - 4903891$ .
Perform Subtraction in Numerator: Perform the subtraction in the numerator: $673456429999999 - 4903891 = 673456425000108$ .
Fraction Representing Number: Now we have the fraction representing the number to be subtracted from $67345643$ : $(673456425000108) / 9999999$ .
Convert Fraction to Decimal: Since we are looking for a whole number to subtract, we need to convert this fraction back into a decimal to see if it represents a whole number. However, we can observe that the numerator is not a multiple of the denominator, which means the fraction does not represent a whole number. This indicates a math error has occurred in the previous steps.