Question

Anne asks the butcher for \(1 \frac{3}{4}\) pounds of ham. The butcher places some ham on the digital scale, which reads \(1.95\). How much ham must the butcher remove to give Anne the amount that she asked for? (Note: You may not use a calculator.) (A) \(0.10\) pounds (B) \(0.20\) pounds (C) \(0.45\) pounds (D) \(0.70\) pounds

Short answer

(B) \(0.20\) pounds

Step-by-step solution

Convert the mixed number to an improper fraction

To convert the mixed number to an improper fraction, follow these steps: 1. Multiply the whole number (1) by the denominator (4). 2. Add the numerator (3) to the result obtained in step 1. 3. Write the result in step 2 as the new numerator with the original denominator.

Convert the improper fraction to a decimal

To convert the improper fraction to a decimal, divide the numerator by the denominator: \(7 / 4 = 1.75\) So, Anne asked for 1.75 pounds of ham.

Subtract Anne's requested amount from the butcher's amount

To find out how much ham the butcher must remove, subtract Anne's requested amount from the butcher's scale amount: \(1.95 - 1.75 = 0.20\) The butcher must remove 0.20 pounds of ham to give Anne the exact amount she asked for.

Match the answer to the given options

The answer we found in step 3 is 0.20, which matches option (B). So, the correct answer is (B) \(0.20\) pounds.

Key concepts

Converting Mixed Numbers

When dealing with numbers that combine whole numbers and fractions, we call them 'mixed numbers.' To convert a mixed number into an improper fraction, follow these simple steps:

• Multiply the whole number by the denominator of the fraction. This brings the whole number into the same form as the fraction.
• Add the numerator of the fraction to your result from the first step. This gives you the total count of equal parts.
• Write this total as the numerator over the original denominator. This is your improper fraction.

For example, in the case of Anne's ham request of \(1 \frac{3}{4}\) pounds, first multiply the whole number 1 by the denominator 4 to get 4. Add the numerator 3 to this product, resulting in 7. Therefore, \(1 \frac{3}{4}\) as an improper fraction is \(\frac{7}{4}\). This method is essential when you need to work with mixed numbers in calculations.

Decimal Subtraction

Once we have converted numbers into decimals, subtraction becomes straightforward. Subtracting decimals follows similar rules to subtracting whole numbers, just remember to line up the decimal points:

• Align the numbers by their decimal points. This ensures that each place value is appropriately matched.
• Proceed to subtract each digit, starting from the rightmost digit (usually the tenths place). Borrow from the next column as needed, just like with whole numbers.
• Place the decimal point directly below the other decimal points in the result.

In the example provided, subtracting the requested amount by Anne, 1.75, from the butcher's amount, 1.95, involves subtracting like places: \(1.95 - 1.75 = 0.20\). This calculated difference tells us exactly how much excess ham the butcher should remove.

Fraction to Decimal Conversion

Converting fractions to decimals is a crucial skill in various mathematical tasks and everyday calculations. When you convert a fraction into a decimal, you are essentially performing a division of the numerator by the denominator:

• Divide the top number (numerator) by the bottom number (denominator). Using long division, try to reach an accurate decimal point; for some fractions, this could reveal a repeating decimal.
• The quotient, which results from the division, is the decimal equivalent of the fraction.
• This process is particularly useful when comparing quantities or performing operations like addition or subtraction with decimal numbers.

In Anne's case, converting \(\frac{7}{4}\) involves dividing 7 by 4. This calculation results in the decimal 1.75, which makes it easier to compare with another decimal such as 1.95. Knowing how to convert between these two representations allows flexibility and precision during problem solving.