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Chapter 1: Problem 18

$$27 \div \frac{9}{5}-1$$

Short Answer

Expert verified
The short answer to the given expression \(27 \div \frac{9}{5}-1\) is: 1. Rewrite the expression as \(27 \cdot \frac{5}{9} - 1\). 2. Multiply: \(27 \cdot \frac{5}{9} = 15\). 3. Subtract: \(15 - 1 = 14\). So, the solution is \(\boxed{14}\).

Step by step solution

01

Rewrite the division as multiplication using reciprocal

To divide 27 by \(\frac{9}{5}\), we first need to rewrite the division as multiplication by using the reciprocal of the fraction. The reciprocal of \(\frac{9}{5}\) is \(\frac{5}{9}\). So, we rewrite the expression as follows: \(27 \div \frac{9}{5}-1 = 27 \cdot \frac{5}{9} - 1\)
02

Multiply 27 by the reciprocal, 5/9

Now, we perform the multiplication. To multiply a whole number by a fraction, you can simply multiply the whole number by the numerator of the fraction and then divide by the denominator: \(27 \cdot \frac{5}{9} = \frac{27 \times 5}{9}\) To simplify this, we can first cancel out the common factor between 27 and 9 to make the calculation easier: \(\frac{27 \times 5}{9} = \frac{3 \times 5}{1} = \frac{15}{1}\) Since 15 divided by 1 is just 15, so: \(27 \cdot \frac{5}{9} = 15\)
03

Subtract 1 from the result

Finally, subtract 1 from the result of the multiplication: \(15 - 1 = 14\) Hence, the solution to the given expression is: \(\boxed{14}\)

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Key Concepts

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

Reciprocal of a Fraction
When dividing by a fraction, it can sometimes seem confusing. But there's a nifty trick to make things easier: use the reciprocal! The **reciprocal** of a fraction is simply flipping the numerator and the denominator. Think of it as turning the fraction upside down. For example:
  • The reciprocal of \( \frac{9}{5} \) is \( \frac{5}{9} \).
  • Generally, if you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This concept is particularly useful because it allows us to change a division problem into a multiplication one. This means, instead of dividing by a fraction, you can multiply by its reciprocal, which is often easier to handle. Next time you see a division with fractions, remember: flip the fraction and multiply!
Multiplication of Fractions
Multiplying fractions might look tricky at first, but it's one of the simpler operations with fractions. Here's why:When you multiply two fractions, you multiply the numerators (the top numbers) together and multiply the denominators (the bottom numbers) together. If you have a whole number and a fraction, you can treat the whole number as a fraction where the denominator is 1.Example:
  • To multiply 27 by \( \frac{5}{9} \), think of 27 as \( \frac{27}{1} \).
  • This means you multiply straight across: \( \frac{27}{1} \times \frac{5}{9} = \frac{135}{9} \).
It鈥檚 helpful to simplify before multiplying to make calculations easier, which brings us to our next topic of simplifying expressions.
Simplifying Expressions
Simplifying mathematical expressions helps in making calculations quick and easy. Once you multiply fractions, the next step is often to simplify the result.Start by looking for common factors. This means finding numbers that appear in both the numerator and denominator and canceling them out. This way, you deal with smaller numbers and clearer outcomes.For instance, look at \( \frac{135}{9} \). Both 135 and 9 have a common factor, which is 9.
  • Divide both the numerator (135) and the denominator (9) by their greatest common factor, 9.
  • This gives \( \frac{135 \div 9}{9 \div 9} = \frac{15}{1} \).
Now, \( \frac{15}{1} \) can be easily understood as just 15. Simplifying not only makes the numbers friendlier but also hones your problem-solving skills to manage larger, complex problems efficiently.

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