Chapter 1 Use Positive Rational Numbers
Section 1.3 Multiply Fractions
Page 19 Exercise 1 Answer
Since the art teacher gave each student half of a sheet of paper and then she asked the students to color one fourth of their pieces of paper, the students colored \(\frac{1}{4}\) of \(\frac{1}{2}\) of the original sheet of paper.
\(\frac{1}{4} \times \frac{1}{2}=\frac{1 \times 1}{4 \times 2}=\frac{1}{8}\)Result
The students colored \(\frac{1}{8}\) of the original sheet of paper.
Page 19 Exercise 1 Answer
Since each student got half of the original paper and then colored only a small piece of it, the answer must be less than one.
The students colored \(\frac{1}{8}\) of the original sheet of paper, which is less than one.
Result
Since each student got half of the original paper and then colored only a small piece of it, the answer must be less than one.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 20 Exercise 1 Answer
To find \(\frac{1}{4}\) x \(\frac{1}{5}\) using an area model, we need to start by dividing a rectangle into 4 rows and 5 columns. Then we shade 1 to the 4 rows to represent \(\frac{1}{4}\) and shade 1 of the 5 columns to represent \(\frac{1}{5}\).
The overlap shows the product \(\frac{1}{4}\) x \(\frac{1}{5}\). Since 1 out of 20 parts are shaded twice, then \(\frac{1}{4}\) x \(\frac{1}{5}\) = \(\frac{1}{20}\)
When multiplying a number by a number smaller than 1, the product is always smaller than the original number. This means that \(\frac{1}{4}\) times a number smaller than 1 will give a product less than \(\frac{1}{4}\). It also means that \(\frac{1}{5}\) times a number smaller than 1 will give a product less than \(\frac{1}{5}\). Therefore, the product \(\frac{1}{4}\) x \(\frac{1}{5}\) must be less than both factors of \(\frac{1}{4}\) and \(\frac{1}{5}\).
Result
1 of 4 rows 1 of 5 columns \(\frac{1}{4}\) x \(\frac{1}{5}\) = \(\frac{1}{20}\)
The product is less than both factors because multiplying a number by a number smaller than 1 always gives a product that is smaller than the original number and \(\frac{1}{4}\) and \(\frac{1}{5}\) are both smaller than 1.
Page 21 Exercise 2 Answer
We need to find \(\frac{3}{4} \times \frac{4}{6}\) using the number line.
\(\frac{1}{4}\) means 1 of 4 equal parts so \(\frac{1}{4}\) of \(\frac{4}{6}\) is \(\frac{1}{4} \times \frac{4}{6}=\frac{4}{24}=\frac{1}{6}\)
Since \(\frac{1}{4}\) of \(\frac{4}{6}\) is \(\frac{1}{6}\), draw arrows from 0 to \(\frac{1}{6}\), from \(\frac{1}{6}\) to \(\frac{2}{6}\), from \(\frac{2}{6}\) to \(\frac{3}{6}\), and from \(\frac{3}{6}\) to \(\frac{4}{6}\). Label each arrow as \(\frac{1}{4}\).
Since \(\frac{3}{4}\) means 3 of 4 equal parts, then \(\frac{3}{4}\) of \(\frac{4}{6}\) is 3 times. \(\frac{1}{6}\). The product is then where the third arrow stopped, which was \(\frac{3}{6}\).
Reducing \(\frac{3}{6}\) gives the final answer of \(\frac{3}{4} \times \frac{4}{6}=\frac{1}{2}\)
Result
\(\frac{1}{2}\)Page 21 Exercise 3 Answer
A clothing factory makes jackets. If each machine makes \(3 \frac{1}{3}\) jackets per hour, how many jackets does one machine make in \(4 \frac{1}{2}\) hours?
\(3 \frac{1}{3} \times 4 \frac{1}{2}=\frac{10}{3} \times \frac{9}{2}=\frac{10 \times 9}{3 \times 2}=\frac{90}{6}=15\)Result
One machine makes 15 jackets in \(4 \frac{1}{2}\).
Page 22 Exercise 1 Answer
To multiply fractions and mixed numbers we must first write mixed numbers as fractions and then simply multiply as we multiply fractions. We multiply the numerators and multiply the denominators. If possible rewrite the result as a mixed number.
For example,
\(\frac{3}{5} \times 3 \frac{2}{8}=\frac{3}{5} \times \frac{26}{8}=\frac{3 \times 26}{5 \times 8}=\frac{78}{40}=\frac{39}{20}=\frac{20}{20}+\frac{19}{20}=1 \frac{19}{20} .\)Result
To multiply fractions and mixed numbers we must first write mixed numbers as fractions and then simply multiply as we multiply fractions. We multiply the numerators and multiply the denominators. If possible rewrite the result as a mixed number.
Page 22 Exercise 2 Answer
\(\frac{3}{6} \times \frac{5}{4}=\frac{3 \times 5}{6 \times 4}=\frac{15}{24}=\frac{5}{8}\) \(\frac{3}{4} \times \frac{5}{6}=\frac{3 \times 5}{4 \times 6}=\frac{15}{24}=\frac{5}{8}\)As we can see the products \(\frac{3}{6}\) x \(\frac{5}{4}\) and \(\frac{3}{4}\) x \(\frac{5}{6}\) are equal. The reason is that multiplying fractions comes down to multiplying the numerators and multiplying the denominators, since both numerators and both denominators are whole numbers all the rules for mulitplication of whole numbers apply, more specifically the commutative rule.
However, the products \(\frac{3}{5}\) x \(\frac{6}{4}\) would not be equal since we switched the numerator for a denominator. But as long as only we switch a numerator with a numerator and a denominator with a denominator, the products will be equal.
Result
Yes, \(\frac{3}{6}\) x \(\frac{5}{4}\) and \(\frac{3}{4}\) x \(\frac{5}{6}\) are equal since they both give \(\frac{15}{24}\) = \(\frac{5}{8}\)
Page 22 Exercise 3 Answer
\(\frac{3}{9}+\frac{6}{9}=\frac{3+6}{9}=\frac{9}{9}=1\)When adding fractions the denominator stays the same, and we only add the numerators. In words, if have three ninths of, let’s say pizza, and add to that six ninths of pizza we have nine ninths or one pizza.
\(\frac{3}{9} \times \frac{6}{9}=\frac{3 \times 6}{9 \times 9}=\frac{18}{81}=\frac{2}{9}\)When multiplying fractions we multiply a numerator by a numerator and a denominator by a denominator.
Result
When adding fractions the denominator stays the same, and we only add the numerators. When multiplying fractions we multiply a numerator by a numerator and a denominator by a denominator.
Page 22 Exercise 4 Answer
Tina ate half, or \(\frac{1}{2}\), of what was left from the dinner party, which is \(\frac{1}{2}\) of a pan of cornbread.
\(\frac{1}{2} \times \frac{1}{2}=\frac{1 \times 1}{2 \times 2}=\frac{1}{4}\)Result
Tina ate \(\frac{1}{4}\) of a pan of cornbread.
Page 22 Exercise 5 Answer
To multiply 5 x 2 \(\frac{1}{2}\) first we must rewrite both numbers as fractions. Multiply the numerators. Multiply the denominators. If needed, rewrite the result as a mixed number.
\(5 \times 2 \frac{1}{2}=\frac{5}{1} \times \frac{5}{2}=\frac{5 \times 5}{1 \times 2}=\frac{25}{2}=\frac{24}{2}+\frac{1}{2}=12 \frac{1}{2}\)Result
Rewrite both numbers as fractions, multiply the numerators, multiply the denominators, and then rewrite the result as a mixed number. The product is then \(5 \times 2 \frac{1}{2}=12 \frac{1}{2}\)
Page 22 Exercise 6 Answer
Tom ate \(\frac{1}{3}\) of the lasagna which was left – which is \(\frac{7}{8}\) of a whole pan. To find the fraction of a whole pan that Tom ate find the product \(\frac{1}{3} \times \frac{7}{8}\)
\(\frac{1}{3} \times \frac{7}{8}=\frac{1 \times 7}{3 \times 8}=\frac{7}{24}\)Result
\(\frac{7}{24}\)Page 22 Exercise 7 Answer
Shade 5 (out of 6) rows in yellow.
Shade 1 (out of 2) columns in red.
Cells overlapped by an orange color represent the product \(\frac{5}{6} \times \frac{1}{2}\)
Notice that 5 out of 12 cells are colored in orange, thus
\(\frac{5}{6} \times \frac{1}{2}=\frac{5}{12}\)Result
\(\frac{5}{12}\)Page 22 Exercise 8 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{3}{4} \times \frac{4}{9}=\frac{3 \times 4}{4 \times 9}=\frac{12}{36}=\frac{1}{3}\)Result
\(\frac{1}{3}\)Page 22 Exercise 9 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{2}{3} \times \frac{1}{2}=\frac{2 \times 1}{3 \times 2}=\frac{2}{6}=\frac{1}{3}\)Result
\(\frac{1}{3}\)Page 22 Exercise 10 Answer
Multiply the numerators and multiply the denominators:
\(\frac{5}{9} \times \frac{1}{9}=\frac{5 \times 1}{9 \times 9}=\frac{5}{81}\)Result
\(\frac{5}{81}\)Page 22 Exercise 11 Answer
Multiply the numerators and multiply the denominators:
\(\frac{7}{10} \times \frac{3}{4}=\frac{7 \times 3}{10 \times 4}=\frac{21}{40}\)Result
\(\frac{21}{40}\)Page 22 Exercise 12 Answer
Multiply the numerators and multiply the denominators:
\(\frac{1}{3} \times \frac{1}{4}=\frac{1 \times 1}{3 \times 4}=\frac{1}{12}\)Result
\(\frac{1}{12}\)Page 22 Exercise 13 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{5}{6} \times \frac{3}{7}=\frac{5 \times 3}{6 \times 7}=\frac{15}{42}=\frac{5}{14}\)Result
\(\frac{5}{14}\)Page 22 Exercise 14 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{3}{5} \times \frac{11}{12}=\frac{3 \times 11}{5 \times 12}=\frac{33}{60}=\frac{11}{20}\)Result
\(\frac{11}{20}\)Page 22 Exercise 15 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{4}{10} \times \frac{2}{5}=\frac{4 \times 2}{10 \times 5}=\frac{8}{50}=\frac{4}{25}\)Result
\(\frac{4}{25}\)Page 22 Exercise 16 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{3}{4} \times \frac{2}{9}=\frac{3 \times 2}{4 \times 9}=\frac{6}{36}=\frac{1}{6}\)Result
\(\frac{1}{6}\)Page 22 Exercise 17 Answer
To estimate the product \(2 \frac{3}{4} \times 8\), we can round mixed number to teh nearest whole number and then multiply:
\(2 \frac{3}{4} \times 8 \approx 3 \times 8=24\)To complete the multiplication, rewrite the numbers as improper fractions. Then multiply the numerators and denominators and reduce the fraction:
\(2 \frac{3}{4} \times 8=\frac{11}{4} \times \frac{8}{1}=\frac{11 \times 8}{4 \times 1}=\frac{88}{4}=22\)Result
Estimate: 24 Completed multiplication:
\(2 \frac{3}{4} \times 8=\frac{11}{4} \times \frac{8}{1}=22\)Page 22 Exercise 18 Answer
To estimate the product \(4 \frac{1}{2} \times 1 \frac{1}{4}\), we can round mixed number to teh nearest whole number and then multiply:
\(4 \frac{1}{2} \times 1 \frac{1}{4} \approx 5 \times 1=5\)To complete the multiplication, rewrite the numbers as improper fractions. Then multiply the numerators and denominators and reduce the fraction:
\(4 \frac{1}{2} \times 1 \frac{1}{4}=\frac{9}{2} \times \frac{5}{4}=\frac{9 \times 5}{2 \times 4}=\frac{45}{8}=5 \frac{5}{8}\)Result
Estimate: 24 Completed multiplication:
\(4 \frac{1}{2} \times 1 \frac{1}{4}=\frac{9}{2} \times \frac{5}{4}=5 \frac{5}{8}\)Page 23 Exercise 19 Answer
We need to find \(\frac{1}{3}\) x \(\frac{5}{6}\) using the given bar model.
The given bar model is already divided up into sixths.
We need to find \(\frac{1}{3}\) of \(\frac{5}{6}\) so divide each of the sixths into 3 equal parts.
Then shade 1 of the 3 equal parts for five of the sixths.
The entire bar model is divided up into 18 parts and 5 of them are shaded. Therefore \(\frac{1}{3} \times \frac{5}{6}=\frac{5}{18}\)
Result
\(\frac{5}{18}\)Page 23 Exercise 20 Answer
Shade 2 (out of 3) rows in yellow.
Shade 1 (out of 12) columns in red.
Cells overlapped by an orange color represent the product \(\frac{2}{3}\) x \(\frac{1}{12}\).
Notice that 2 out of 26 cells are colored in orange, thus
\(\frac{2}{3} \times \frac{1}{12}=\frac{2}{36} \text { or } \frac{1}{18}\)Result
\(\frac{2}{36} \text { or } \frac{1}{18}\)Page 23 Exercise 21 Answer
Multiply the numerators and multiply the denominators:
\(\frac{7}{8} \times \frac{1}{2}=\frac{7 \times 1}{8 \times 2}=\frac{7}{16}\)Result
\(\frac{7}{16}\)Page 23 Exercise 22 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{2}{5} \times \frac{1}{12}=\frac{2 \times 1}{5 \times 12}=\frac{2}{60}=\frac{1}{30}\)Result
\(\frac{1}{30}\)Page 23 Exercise 23 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{5}{7} \times \frac{7}{9}=\frac{5 \times 7}{7 \times 9}=\frac{35}{63}=\frac{5}{9}\)Result
\(\frac{5}{9}\)Page 23 Exercise 24 Answer
Multiply the numerators and multiply the denominators:
\(\frac{1}{2} \times \frac{3}{4}=\frac{1 \times 3}{2 \times 4}=\frac{3}{8}\)Result
\(\frac{3}{8}\)Page 23 Exercise 25 Answer
Multiply the numerators and multiply the denominators:
\(\frac{1}{4} \times \frac{7}{8}=\frac{1 \times 7}{4 \times 8}=\frac{7}{32}\)Result
\(\frac{7}{32}\)Page 23 Exercise 26 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{5}{6} \times \frac{9}{10}=\frac{5 \times 9}{6 \times 10}=\frac{45}{60}=\frac{3}{4}\)Result
\(\frac{3}{4}\)Page 23 Exercise 27 Answer
Multiply the numerators and multiply the denominators:
\(\frac{1}{4} \times \frac{1}{8}=\frac{1 \times 1}{4 \times 8}=\frac{1}{32}\)Result
\(\frac{1}{32}\)Page 23 Exercise 26 Answer
Multiply the numerators, multiply the denominators, and then reduce the fraction:
\(\frac{1}{3} \times \frac{3}{7}=\frac{1 \times 3}{3 \times 7}=\frac{3}{21}=\frac{1}{7}\)Result
\(\frac{1}{7}\)Page 23 Exercise 27 Answer
We need to estimate and find the actual product \(2 \frac{1}{6}\) x \(4 \frac{1}{2}\).
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(2 \frac{1}{6} \times 4 \frac{1}{2} \approx 2 \times 5=10\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(2 \frac{1}{6} \times 4 \frac{1}{2}=\frac{13}{6} \times \frac{9}{2}=\frac{13 \times 9}{6 \times 2}=\frac{117}{12}=\frac{39}{4}=9 \frac{3}{4}\)Result
Estimate: 10
Actual: \(9 \frac{3}{4}\)
Page 23 Exercise 30 Answer
We need to estimate and find the actual product \(\frac{3}{4}\) x \(8 \frac{1}{2}\).
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(\frac{3}{4} \times 8 \frac{1}{2} \approx \frac{3}{4} \times 8=\frac{24}{4}=6\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(\frac{3}{4} \times 8 \frac{1}{2}=\frac{3}{4} \times \frac{17}{2}=\frac{3 \times 17}{4 \times 2}=\frac{51}{8}=6 \frac{3}{8}\)Result
Estimate: 6
Actual: \(6 \frac{3}{8}\)
Page 23 Exercise 31 Answer
We need to estimate and find the actual product \(1 \frac{1}{8}\) x \(3 \frac{1}{3}\).
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(1 \frac{1}{8} \times 3 \frac{1}{3} \approx 1 \times 3=3\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(1 \frac{1}{8} \times 3 \frac{1}{3}=\frac{9}{8} \times \frac{10}{3}=\frac{9 \times 10}{8 \times 3}=\frac{90}{24}=\frac{15}{4}=3 \frac{3}{4}\)Result
Estimate: 3
Actual: \(3 \frac{3}{4}\)
Page 23 Exercise 32 Answer
We need to estimate and find the actual product \(3 \frac{1}{5}\) x \(\frac{2}{3}\).
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(3 \frac{1}{5} \times \frac{2}{3} \approx 3 \times \frac{2}{3}=\frac{6}{3}=2\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(3 \frac{1}{5} \times \frac{2}{3}=\frac{16}{5} \times \frac{2}{3}=\frac{16 \times 2}{5 \times 3}=\frac{32}{15}=2 \frac{2}{15}\)Result
Estimate: 2
Actual: \(2 \frac{2}{15}\)
Page 23 Exercise 33 Answer
We need to estimate and find the actual product \(3 \frac{1}{4}\) x 6
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(3 \frac{1}{4} \times 6 \approx 3 \times 6=18\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(3 \frac{1}{4} \times 6=\frac{13}{4} \times \frac{6}{1}=\frac{13 \times 6}{4 \times 1}=\frac{78}{4}=\frac{39}{2}=19 \frac{1}{2}\)Result
Estimate: 18
Actual: \(19 \frac{1}{2}\)
Page 23 Exercise 34 Answer
We need to estimate and find the actual product \(5 \frac{1}{3}\) x 3.
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(5 \frac{1}{3} \times 3 \approx 5 \times 3=15\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(5 \frac{1}{3} \times 3=\frac{16}{3} \times \frac{3}{1}=\frac{16 \times 3}{3 \times 1}=\frac{48}{3}=16\)Result
Estimate: 15
Actual: 16
Page 23 Exercise 35 Answer
We need to estimate and find the actual product \(2 \frac{3}{8}\) x 3.
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(2 \frac{3}{8} \times 4 \approx 2 \times 4=8\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(2 \frac{3}{8} \times 4=\frac{19}{8} \times \frac{4}{1}=\frac{19 \times 4}{8 \times 1}=\frac{76}{8}=\frac{19}{2}=9 \frac{1}{2}\)Result
Estimate: 8
Actual: \(9 \frac{1}{2}\)
Page 23 Exercise 36 Answer
We need to estimate and find the actual product \(4 \frac{1}{8}\) x \(5 \frac{1}{2}\).
To find the estimate, round each mixed number to the nearest whole number and then multiply:
\(4 \frac{1}{8} \times 5 \frac{1}{2} \approx 4 \times 6=24\)To find the actual product, rewrite each mixed number as an improper fraction. Multiply the numerators and multiply the denominators. Reduce the fraction and convert to a mixed number if necessary:
\(4 \frac{1}{8} \times 5 \frac{1}{2}=\frac{33}{8} \times \frac{11}{2}=\frac{33 \times 11}{8 \times 2}=\frac{363}{16}=22 \frac{11}{16}\)Result
Estimate: 24
Actual: \(22 \frac{11}{16}\)
Page 23 Exercise 37 Answer
A map is given were we can see three trails, Tremont Trail which is \(3 \frac{1}{2}\) miles long, Seton Trail which is \(1 \frac{1}{4}\) miles long, and Wildflower Trail which is \(2 \frac{3}{8}\) miles long. Linda walked \(\frac{3}{4}\) of the length of the Tremont Trail before stopping for a rest.
\(\frac{3}{4} \times 3 \frac{1}{2}=\frac{3}{4} \times \frac{7}{2}=\frac{3 \times 7}{4 \times 2}=\frac{21}{8}=\frac{16}{8}+\frac{5}{8}=2 \frac{5}{8}\)Linda walked \(2 \frac{5}{8}\) miles before stopping for a rest.
Result
Linda walked \(2 \frac{5}{8}\) miles before stopping for a rest.
Page 23 Exercise 38 Answer
A map is given were we can see three trails, Tremont Trail which is \(3 \frac{1}{2}\) miles long, Seton Trail which is \(1 \frac{1}{4}\) miles long, and Wildflower Trail which is \(2 \frac{3}{8}\) miles long. The city plans to extend the Wildflower Trail to make it \(2 \frac{1}{2}\) times its current length in the next five years.
The current length of the Wildflower Trail is \(2 \frac{3}{8}\) miles.
\(2 \frac{3}{8} \times 2 \frac{1}{2}=\frac{19}{8} \times \frac{5}{2}=\frac{19 \times 5}{8 \times 2}=\frac{95}{16}=\frac{80}{16}+\frac{15}{16}=5 \frac{15}{16}\)At the end of 5 years the Wildflower Trail will be \(5 \frac{15}{16}\) miles long.
Result
At the end of 5 years the Wildflower Trail will be \(5 \frac{15}{16}\) miles long.
Page 24 Exercise 39 Answer
The world’s smalles gecko is \(\frac{3}{4}\) inch long. An adult male Western Banded Gecko is \(7 \frac{1}{3}\) times as long.
\(\frac{3}{4} \times 7 \frac{1}{3}=\frac{3}{4} \times \frac{22}{3}=\frac{3 \times 22}{4 \times 3}=\frac{66}{12}=\frac{11}{2}=\frac{10}{2}+\frac{1}{2}=5 \frac{1}{2}\)Result
An adult male western Banded Gecko is \(5 \frac{1}{2}\) inches long.
Page 24 Exercise 40 Answer
In Ms. Barclay’s classroom, \(\frac{2}{5}\) of the students play chess. There are 30 students in Ms. Barclay’s class.
\(30 \times \frac{2}{5}=\frac{30}{1} \times \frac{2}{5}=\frac{30 \times 2}{1 \times 5}=\frac{60}{5}=12\)12 students in Ms. Barclay’s class play chess. \(\frac{5}{6}\) of these students also play sudoku.
\(12 \times \frac{5}{6}=\frac{12}{1} \times \frac{5}{6}=\frac{12 \times 5}{1 \times 6}=\frac{60}{6}=10\)10 students in Ms. Barclay’s class play chess and sudoku.
Result
10 students play chess and sudoku.
Page 24 Exercise 41 Answer
The Akashi-Kaiko Bridge in Japan is about \(1 \frac{4}{9}\) times as long as the Golden Gate Bridge in San Francisco, which is about 9,000 feet long.
\(9000 \times 1 \frac{4}{9}=\frac{9000}{1} \times \frac{13}{9}=\frac{9000 \times 13}{1 \times 9}=\frac{117000}{9}=\frac{117000}{9}=13000\)Result
The Akashi-Kaiko Bridge is about 13000 feet long.
Page 24 Exercise 42 Answer
Multiplication can be described as a process of scaling, for example, if a 4 inch length is multiplied by two, it is scaled up to 8 inches or twice its original length. If a 4 inch length is multiplied by \(\frac{1}{2}\), it is scaled down to 2 inches or half of its original length.
When a whole number is multiplied by a fraction less than one, the product is less than the whole number, and when a whole number is multiplied by a fraction greater than one, the product is greater than the whole number.
When \(\frac{7}{8}\) and \(\frac{4}{5}\) are multiplied the product will be less than both factors since both of them are less than one.
\(\frac{7}{8} \times \frac{4}{5}=\frac{7 \times 4}{8 \times 5}=\frac{28}{40}=\frac{7}{10}=0.7\)While \(\frac{7}{8}\) = 0.875
\(\frac{4}{5}\) = 0.8
Result
No, the product will be less than both factors.
Page 24 Exercise 43 Answer
To amend the U.S. Constitution, \(\frac{3}{4}\) of the 50 states must approve the amendment. If 35 states approve an amendment, will the Constitution be amended?
\(\frac{3}{4} \times 50=\frac{3}{4} \times \frac{50}{1}=\frac{3 \times 50}{4 \times 1}=\frac{150}{4}=\frac{75}{2}=37.5\)\(\frac{3}{4}\) of 50 is exactly 37.5, so for the Constitution to be amended at least 38 states must approve the amndment. Since only 35 states have, the Constitution will not be amended.
Result
No. If only 35 states approve an amendment, the Constitution won’t be amended.
Page 24 Exercise 44 Answer
A scientist had \(\frac{3}{4}\) of a bottle of solution. She used \(\frac{1}{6}\) of the solution in an experiment.
\(\frac{3}{4} \times \frac{1}{6}=\frac{3 \times 1}{4 \times 6}=\frac{3}{24}=\frac{1}{8}\)In the experiment she used \(\frac{1}{8}\) of the bottle.
Result
She used \(\frac{1}{8}\) of the bottle.
Page 24 Exercise 45 Answer
In the voting for City Council Precinct 5, only \(\frac{1}{2}\) of all eligible voters cast votes. There are two candidates, Shelley and Morgan. Shelly received \(\frac{3}{10}\) of votes and Morgan received \(\frac{5}{8}\) of votes.
To find how many votes Shelly received, we multiply \(\frac{1}{2}\) by \(\frac{3}{10}\) , because \(\frac{1}{2}\) or one half of all eligible voters cast votes and out of them \(\frac{3}{10}\) voted for Shelly.
\(\frac{1}{2} \times \frac{3}{10}=\frac{1 \times 3}{2 \times 10}=\frac{3}{20}\)To find how many votes Morgan received, we multiply \(\frac{1}{2}\) by \(\frac{5}{8}\), because \(\frac{1}{2}\) or one half of all eligible voters cast votes and out of them \(\frac{5}{8}\) voted for Morgan.
\(\frac{1}{2} \times \frac{5}{8}=\frac{1 \times 5}{2 \times 8}=\frac{5}{16}\)Shelly received \(\frac{3}{20}\) votes from all eligible voters and Morgan received \(\frac{5}{16}\) votes.
The find who recieved the most votes we must find the common denominator, which in this case is the least common multiple of 20 and 16.
\(20=2 \times 10=2 \times 2 \times 5=2^2 \times 5\) \(16=2 \times 8=2 \times 2 \times 4=2 \times 2 \times 2 \times 2=2^4\)The LCM is the product of the highest power of each prime number, so 24 x 5, which is 80.
We write \(\frac{3}{20}\) amd \(\frac{5}{16}\) as fractions with the denominator 80.
80 can be written as 20 x 4 as 16 x 5, so we have the following:
\(\frac{3}{20}=\frac{3 \times 4}{20 \times 4}=\frac{12}{80}\) \(\frac{5}{16}=\frac{5 \times 5}{16 \times 5}=\frac{25}{80}\)Shelly received \(\frac{12}{80}\) votes and Morgan received \(\frac{25}{80}\) votes so Morgan received more votes than Shelly.
Result
Shelly received \(\frac{12}{80}\) votes and Morgan received \(\frac{25}{80}\) votes so Morgan received more votes than Shelly.
Page 24 Exercise 46 Answer
Three equations are given and we must select all that are true.
1. \(4 \frac{1}{12} \times 2 \frac{3}{4}=11 \frac{11}{48}\)
\(4 \frac{1}{12} \times 2 \frac{3}{4}=\frac{49}{12} \times \frac{11}{8}=\frac{49 \times 11}{12 \times 8}=\frac{539}{96}=6 \frac{11}{48}\)The first equation is true.
2. \(5 \frac{1}{2} \times 5=25 \frac{1}{2}\)
\(5 \frac{1}{2} \times 5=\frac{11}{2} \times \frac{5}{1}=\frac{11 \times 5}{2 \times 1}=\frac{55}{2}=\frac{54}{2}+\frac{1}{2}=27 \frac{1}{2} \neq 25 \frac{1}{2}\)The second equation is not true.
3. \(2 \frac{1}{5} \times 6 \frac{1}{4}=13 \frac{3}{4}\)
\(2 \frac{1}{5} \times 6 \frac{1}{4}=\frac{11}{5} \times \frac{25}{4}=\frac{11 \times 25}{5 \times 4}=\frac{275}{20}=\frac{260}{20}+\frac{15}{20}=13 \frac{3}{4}\)The third equation is true.
Result
The first, third equations are true.
Page 24 Exercise 47 Answer
Five expressions are given and we have to select all that have \(\frac{3}{4}\) as a product.
\(\frac{1}{2} \times \frac{1}{2}=\frac{1 \times 1}{2 \times 2}=\frac{1}{4} \neq \frac{3}{4}\) \(\frac{9}{10} \times \frac{5}{6}=\frac{9 \times 5}{10 \times 6}=\frac{45}{60}=\frac{3}{4}\) \(\frac{7}{8} \times \frac{6}{7}=\frac{7 \times 6}{8 \times 7}=\frac{42}{56}=\frac{3}{4}\) \(\frac{3}{4} \times \frac{3}{4}=\frac{3 \times 3}{4 \times 4}=\frac{9}{16} \neq \frac{3}{4}\) \(\frac{1}{4} \times \frac{1}{2}=\frac{1 \times 1}{4 \times 2}=\frac{1}{8} \neq \frac{3}{4}\)Result
\(\frac{9}{10} \times \frac{5}{6} \text { and } \frac{7}{8} \times \frac{6}{7}\)