Question

learning goal 2.9: students will fluently multiply and divide multi-digit decimals to the thousandths and multiply and divide positive fractions. 3. $1\frac{17}{30} \times 2\frac{5}{8}$ 4. $(9.58)(6.02)$ 5. $1\frac{21}{25} \div 4$ 6. $52.716 \div 0.092$

Explanation:

Problem 3: \(1\frac{17}{30} \times 2\frac{5}{8}\)

Step 1: Convert mixed numbers to improper fractions

To convert a mixed number \(a\frac{b}{c}\) to an improper fraction, we use the formula \(\frac{a \times c + b}{c}\).
For \(1\frac{17}{30}\):
\(1 \times 30 + 17 = 47\), so \(1\frac{17}{30} = \frac{47}{30}\).

For \(2\frac{5}{8}\):
\(2 \times 8 + 5 = 21\), so \(2\frac{5}{8} = \frac{21}{8}\).

Step 2: Multiply the improper fractions

When multiplying fractions \(\frac{n_1}{d_1} \times \frac{n_2}{d_2}\), we multiply the numerators and denominators:
\(\frac{47}{30} \times \frac{21}{8} = \frac{47 \times 21}{30 \times 8}\).

Calculate the numerator: \(47 \times 21 = 987\).
Calculate the denominator: \(30 \times 8 = 240\).

So we have \(\frac{987}{240}\).

Step 3: Simplify the fraction (optional)

We can simplify \(\frac{987}{240}\) by dividing numerator and denominator by their greatest common divisor (GCD). The GCD of 987 and 240 is 3.

Divide numerator: \(987 \div 3 = 329\).
Divide denominator: \(240 \div 3 = 80\).

So \(\frac{987}{240} = \frac{329}{80}\), which can also be written as a mixed number: \(4\frac{9}{80}\) (since \(329 \div 80 = 4\) with a remainder of 9).

Problem 4: \((9.58)(6.02)\)

Step 1: Multiply the decimals as if they were whole numbers

First, ignore the decimal points and multiply 958 by 602.

\(958 \times 602\):
We can expand this as \(958 \times (600 + 2) = 958 \times 600 + 958 \times 2\).

Calculate \(958 \times 600 = 574800\).
Calculate \(958 \times 2 = 1916\).

Add the results: \(574800 + 1916 = 576716\).

Step 2: Determine the number of decimal places

The number \(9.58\) has 2 decimal places, and \(6.02\) has 2 decimal places. So the total number of decimal places in the product is \(2 + 2 = 4\).

Step 3: Place the decimal point

Starting from the right of 576716, move the decimal point 4 places to the left:
\(576716
ightarrow 57.6716\) (since we need 4 decimal places: \(57.6716\)).

Problem 5: \(1\frac{21}{25} \div 4\)

Step 1: Convert the mixed number to an improper fraction

Using the formula for converting mixed numbers to improper fractions:
For \(1\frac{21}{25}\):
\(1 \times 25 + 21 = 46\), so \(1\frac{21}{25} = \frac{46}{25}\).

Step 2: Divide by the whole number (convert to multiplication by reciprocal)

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 4 is \(\frac{1}{4}\).

So \(\frac{46}{25} \div 4 = \frac{46}{25} \times \frac{1}{4}\).

Step 3: Multiply the fractions

Multiply the numerators and denominators:
\(\frac{46 \times 1}{25 \times 4} = \frac{46}{100}\).

Step 4: Simplify the fraction

Simplify \(\frac{46}{100}\) by dividing numerator and denominator by their GCD (which is 2):
\(\frac{46 \div 2}{100 \div 2} = \frac{23}{50}\).
As a decimal, \(\frac{23}{50} = 0.46\).

Problem 6: \(52.716 \div 0.092\)

Answer:

s:

1. \(1\frac{17}{30} \times 2\frac{5}{8} = \boldsymbol{4\frac{9}{80}}\) (or \(\boldsymbol{\frac{329}{80}}\))
2. \((9.58)(6.02) = \boldsymbol{57.6716}\)
3. \(1\frac{21}{25} \div 4 = \boldsymbol{\frac{23}{50}}\) (or \(\boldsymbol{0.46}\))
4. \(52.716 \div 0.092 = \boldsymbol{573}\)