a) Write the number 8488456019500 in words. b) Evaluate the expression $1.0\overline{3} + 2.\overline{428571}$ and leave your answer in fractional form. c) The sum of two numbers i... a) Write the number 8488456019500 in words. b) Evaluate the expression $1.0\overline{3} + 2.\overline{428571}$ and leave your answer in fractional form. c) The sum of two numbers is twenty-three. Twice the larger number is four more than four times the smaller. What are the numbers?

Understand the Problem

There are three sub-questions. The first sub-question asks to write a given numerical amount of money in words. The second sub-question requires one to evaluate a mathematical expression involving repeating decimals and present the solution in fractional form. The third sub-question presents a word problem involving two numbers with a given sum. The relationship between the numbers is further detailed, allowing the formation of a system of linear equations to solve for the two numbers.

Answer

(a) Eight trillion, four hundred eighty-eight billion, four hundred fifty-six million, nineteen thousand, five hundred. (b) $\frac{21}{5}$ (c) 16 and 7

Steps to Solve

Write the amount in words

The number 8,488,456,019,500 is written as: Eight trillion, four hundred eighty-eight billion, four hundred fifty-six million, nineteen thousand, five hundred.

Convert the first repeating decimal to a fraction

Let $x = 1.0\overline{3} = 1.03333...$.

Multiply by 10: $10x = 10.3333...$

Multiply by 100: $100x = 103.3333...$

Subtract the two equations: $100x - 10x = 103.3333... - 10.3333...$, which simplifies to $90x = 93$.

Solve for $x$: $x = \frac{93}{90} = \frac{31}{30}$.

Convert the second repeating decimal to a fraction

Let $y = 2.\overline{428571} = 2.428571428571...$. The repeating block has 6 digits.

Multiply by $10^6 = 1,000,000$: $1000000y = 2428571.428571428571...$

Subtract the two equations: $1000000y - y = 2428571.428571... - 2.428571...$, which simplifies to $999999y = 2428569$.

Solve for $y$: $y = \frac{2428569}{999999} = \frac{809523}{333333} = \frac{27}{26}$.

Add the two fractions

$1.0\overline{3}+2.\overline{428571} = \frac{31}{30} + \frac{19}{6} = \frac{31}{30} + \frac{95}{30} = \frac{126}{30}$.

Simplify the fraction: $\frac{126}{30} = \frac{63}{15} = \frac{21}{5}$.

Set up equations for the word problem

Let $x$ be the larger number and $y$ be the smaller number.

From the problem, we have two equations: $x + y = 23$ $2x = 4y + 4$

Solve the system of linear equations

Solve the first equation for $x$: $x = 23 - y$.

Substitute this into the second equation: $2(23 - y) = 4y + 4$.

Simplify and solve for $y$: $46 - 2y = 4y + 4 \implies 6y = 42 \implies y = 7$.

Substitute $y = 7$ back into $x = 23 - y$ to find $x$: $x = 23 - 7 = 16$.

(a) Eight trillion, four hundred eighty-eight billion, four hundred fifty-six million, nineteen thousand, five hundred. (b) $\frac{21}{5}$ (c) The numbers are 16 and 7.

More Information

The repeating decimal $0.\overline{428571}$ is equal to $\frac{3}{7}$, so $2.\overline{428571} = 2 + \frac{3}{7} = \frac{17}{7}$.

Tips

For part (a), a common mistake is miscounting the place values when writing the number in words. One way to avoid this is to break the number into groups of three digits (from right to left) and name each group (thousands, millions, billions, trillions).
For part (b), a common mistake is incorrectly converting the repeating decimals into fractions. It's important to identify the repeating block and set up the equations correctly.
For part (c), a common mistake is to incorrectly set up the equations based on the word problem. Careful reading and translating the words into mathematical expressions is crucial.

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