Decimal Representation Problem: Solving For B,ac

by Admin 49 views
Decimal Representation Problem: Solving for b,ac

Hey guys! Let's dive into a cool math problem today that involves understanding decimal representations of rational numbers. This problem might seem a bit tricky at first, but don't worry, we'll break it down step by step. We're going to figure out how to find a rational number when we're given information about the decimal representation of another related rational number. So, buckle up and let's get started!

Understanding the Problem

The problem states that the decimal representation of the rational number 8 14/25 is a,bc. This means that when we convert the mixed number 8 14/25 into a decimal, the result will be a decimal number where 'a' is the whole number part, 'b' is the tenths digit, and 'c' is the hundredths digit. Our mission, should we choose to accept it (and we do!), is to find the rational number whose decimal representation is b,ac. This means we need to swap the digits 'b' and 'a' and keep 'c' in the hundredths place. To successfully navigate this mathematical quest, we'll need to convert the mixed number to a decimal, identify the digits a, b, and c, then construct the new decimal b,ac, and finally, convert it back to a rational number. This journey involves fractions, decimals, and a bit of algebraic thinking – the trifecta of mathematical fun!

Converting 8 14/25 to Decimal Form

First, let's convert the mixed number 8 14/25 into a decimal. To do this, we need to focus on the fractional part, which is 14/25. We want to convert this fraction into a decimal. The easiest way to do this is to make the denominator a power of 10 (like 10, 100, 1000, etc.). Since 25 can be easily multiplied to get 100, we'll multiply both the numerator and the denominator of 14/25 by 4. This gives us (14 * 4) / (25 * 4) = 56/100. Now, 56/100 can be directly written as the decimal 0.56. Remember, fractions with denominators that are powers of 10 are super easy to convert to decimals because the denominator tells us the place value of the last digit in the numerator. For example, if the denominator is 100, the last digit is in the hundredths place. Adding this decimal part to the whole number part, which is 8, we get 8 + 0.56 = 8.56. Ta-da! We've successfully converted 8 14/25 to its decimal representation.

Identifying a, b, and c

Now that we have 8.56 as the decimal representation, we can easily identify the values of a, b, and c. Remember, 'a' is the whole number part, 'b' is the tenths digit, and 'c' is the hundredths digit. So, looking at 8.56, we can see that:

  • a = 8 (the whole number part)
  • b = 5 (the tenths digit)
  • c = 6 (the hundredths digit)

These are the building blocks we need to construct our new decimal representation. It's like having the key ingredients for a recipe – now we just need to put them together in the right order!

Constructing b,ac and Converting to Rational Form

Okay, now for the fun part! We know that a = 8, b = 5, and c = 6. The problem asks us to find the rational number whose decimal representation is b,ac. This means we need to swap 'a' and 'b' and keep 'c' in the hundredths place. So, b,ac becomes 5.86. Now, our goal is to convert this decimal, 5.86, back into a rational number. Converting decimals to fractions is a fundamental skill in math, and it's super useful for problems like this. To convert 5.86 to a fraction, we first write it as a fraction with a denominator of 100 because there are two digits after the decimal point (86). So, 5.86 becomes 586/100. But we're not done yet! We need to simplify this fraction to its lowest terms. Both 586 and 100 are even numbers, so they are divisible by 2. Dividing both the numerator and the denominator by 2, we get 293/50. Can we simplify it further? Let's see... 293 is not divisible by 2 or 5, and 50 can only be divided by 2 and 5. It turns out that 293 is a prime number, so we can't simplify the fraction any further. Therefore, the rational number whose decimal representation is b,ac (which is 5.86) is 293/50. We've successfully converted our new decimal back to a rational number, and that's a major win!

Detailed Solution Breakdown

Let's recap the entire solution step-by-step to make sure we've got it all down. Sometimes, seeing the whole picture helps to solidify our understanding. We started with the rational number 8 14/25 and the information that its decimal representation is a,bc. Our ultimate goal was to find the rational number whose decimal representation is b,ac. Here’s how we tackled it:

  1. Convert 8 14/25 to Decimal:
    • We converted the fractional part, 14/25, to a decimal by multiplying both the numerator and the denominator by 4 to get 56/100.
    • This gave us 0.56 as the decimal representation of 14/25.
    • Adding the whole number part (8), we got 8.56 as the decimal representation of 8 14/25.
  2. Identify a, b, and c:
    • From the decimal 8.56, we identified a = 8 (whole number), b = 5 (tenths digit), and c = 6 (hundredths digit).
  3. Construct b,ac:
    • We swapped 'a' and 'b' to get the new decimal representation b,ac, which is 5.86.
  4. Convert 5.86 to Rational Form:
    • We wrote 5.86 as a fraction with a denominator of 100: 586/100.
    • We simplified the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gave us 293/50.
    • Since 293 is a prime number, we couldn't simplify the fraction any further. So, 293/50 is the rational number we were looking for.

So, we've successfully found the rational number whose decimal representation is b,ac. It's like solving a puzzle, isn't it? Each step builds upon the previous one, and when we put it all together, we get the final answer. Understanding these steps is crucial for tackling similar problems in the future. Math is all about building on concepts, so the more you practice, the better you'll get!

Why This Problem Matters

You might be wondering,