Answer :
Below is a detailed step-by-step solution.
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1. Add
[tex]$$\frac{3}{5}+\frac{1}{3}.$$[/tex]
To add the fractions, we first find a common denominator. The least common denominator of 5 and 3 is 15. Rewrite each fraction:
[tex]$$\frac{3}{5}=\frac{3\times3}{5\times3}=\frac{9}{15}, \qquad \frac{1}{3}=\frac{1\times5}{3\times5}=\frac{5}{15}.$$[/tex]
Now add the numerators:
[tex]$$\frac{9}{15}+\frac{5}{15}=\frac{9+5}{15}=\frac{14}{15}.$$[/tex]
The sum is [tex]$\displaystyle \frac{14}{15}$[/tex], which is approximately [tex]$0.93333\ldots$[/tex].
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2. Order the fractions
[tex]$$\frac{1}{5},\; \frac{5}{10},\; \frac{18}{20},\; \frac{7}{10},\; \frac{4}{5}$$[/tex]
in ascending order.
First, simplify and convert them to their simplest forms or decimal equivalents:
- [tex]$$\frac{1}{5}$$[/tex] remains as is. Its decimal equivalent is [tex]$0.2$[/tex].
- [tex]$$\frac{5}{10}$$[/tex] simplifies to [tex]$$\frac{1}{2}$$[/tex], equivalent to [tex]$0.5$[/tex].
- [tex]$$\frac{18}{20}$$[/tex] simplifies by dividing numerator and denominator by [tex]$2$[/tex] to [tex]$$\frac{9}{10}$$[/tex], which is [tex]$0.9$[/tex].
- [tex]$$\frac{7}{10}$$[/tex] remains as is, which is [tex]$0.7$[/tex].
- [tex]$$\frac{4}{5}$$[/tex] remains as is, which is [tex]$0.8$[/tex].
Now, list the decimal values in ascending order:
[tex]$$0.2,\quad 0.5,\quad 0.7,\quad 0.8,\quad 0.9.$$[/tex]
These correspond to:
[tex]$$\frac{1}{5},\quad \frac{1}{2},\quad \frac{7}{10},\quad \frac{4}{5},\quad \frac{9}{10}.$$[/tex]
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3. Solve the proportion
[tex]$$\frac{3}{14}=\frac{x}{70}.$$[/tex]
To find [tex]$x$[/tex], cross-multiply:
[tex]$$14x=3\times70.$$[/tex]
Calculating the right side:
[tex]$$3 \times 70 = 210.$$[/tex]
Now solve for [tex]$x$[/tex]:
[tex]$$x=\frac{210}{14}=15.$$[/tex]
Thus, [tex]$x=15$[/tex].
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4. Subtract
[tex]$$\frac{3}{4}-\frac{1}{3}.$$[/tex]
Find a common denominator; the least common denominator for [tex]$4$[/tex] and [tex]$3$[/tex] is [tex]$12$[/tex]. Rewrite each fraction:
[tex]$$\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}, \qquad \frac{1}{3}=\frac{1\times4}{3\times4}=\frac{4}{12}.$$[/tex]
Now perform the subtraction:
[tex]$$\frac{9}{12}-\frac{4}{12}=\frac{9-4}{12}=\frac{5}{12}.$$[/tex]
The result is [tex]$\displaystyle \frac{5}{12}$[/tex], which is approximately [tex]$0.41667\ldots$[/tex].
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Final Answers:
1. [tex]$$\frac{3}{5}+\frac{1}{3}=\frac{14}{15}\qquad (\approx 0.93333)$$[/tex]
2. The ascending order is: [tex]$$\frac{1}{5},\quad \frac{1}{2},\quad \frac{7}{10},\quad \frac{4}{5},\quad \frac{9}{10}.$$[/tex]
3. [tex]$$\frac{3}{14}=\frac{15}{70}\quad \text{(i.e., }x=15\text{)}.$$[/tex]
4. [tex]$$\frac{3}{4}-\frac{1}{3}=\frac{5}{12}\qquad (\approx 0.41667).$$[/tex]
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1. Add
[tex]$$\frac{3}{5}+\frac{1}{3}.$$[/tex]
To add the fractions, we first find a common denominator. The least common denominator of 5 and 3 is 15. Rewrite each fraction:
[tex]$$\frac{3}{5}=\frac{3\times3}{5\times3}=\frac{9}{15}, \qquad \frac{1}{3}=\frac{1\times5}{3\times5}=\frac{5}{15}.$$[/tex]
Now add the numerators:
[tex]$$\frac{9}{15}+\frac{5}{15}=\frac{9+5}{15}=\frac{14}{15}.$$[/tex]
The sum is [tex]$\displaystyle \frac{14}{15}$[/tex], which is approximately [tex]$0.93333\ldots$[/tex].
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2. Order the fractions
[tex]$$\frac{1}{5},\; \frac{5}{10},\; \frac{18}{20},\; \frac{7}{10},\; \frac{4}{5}$$[/tex]
in ascending order.
First, simplify and convert them to their simplest forms or decimal equivalents:
- [tex]$$\frac{1}{5}$$[/tex] remains as is. Its decimal equivalent is [tex]$0.2$[/tex].
- [tex]$$\frac{5}{10}$$[/tex] simplifies to [tex]$$\frac{1}{2}$$[/tex], equivalent to [tex]$0.5$[/tex].
- [tex]$$\frac{18}{20}$$[/tex] simplifies by dividing numerator and denominator by [tex]$2$[/tex] to [tex]$$\frac{9}{10}$$[/tex], which is [tex]$0.9$[/tex].
- [tex]$$\frac{7}{10}$$[/tex] remains as is, which is [tex]$0.7$[/tex].
- [tex]$$\frac{4}{5}$$[/tex] remains as is, which is [tex]$0.8$[/tex].
Now, list the decimal values in ascending order:
[tex]$$0.2,\quad 0.5,\quad 0.7,\quad 0.8,\quad 0.9.$$[/tex]
These correspond to:
[tex]$$\frac{1}{5},\quad \frac{1}{2},\quad \frac{7}{10},\quad \frac{4}{5},\quad \frac{9}{10}.$$[/tex]
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3. Solve the proportion
[tex]$$\frac{3}{14}=\frac{x}{70}.$$[/tex]
To find [tex]$x$[/tex], cross-multiply:
[tex]$$14x=3\times70.$$[/tex]
Calculating the right side:
[tex]$$3 \times 70 = 210.$$[/tex]
Now solve for [tex]$x$[/tex]:
[tex]$$x=\frac{210}{14}=15.$$[/tex]
Thus, [tex]$x=15$[/tex].
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4. Subtract
[tex]$$\frac{3}{4}-\frac{1}{3}.$$[/tex]
Find a common denominator; the least common denominator for [tex]$4$[/tex] and [tex]$3$[/tex] is [tex]$12$[/tex]. Rewrite each fraction:
[tex]$$\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}, \qquad \frac{1}{3}=\frac{1\times4}{3\times4}=\frac{4}{12}.$$[/tex]
Now perform the subtraction:
[tex]$$\frac{9}{12}-\frac{4}{12}=\frac{9-4}{12}=\frac{5}{12}.$$[/tex]
The result is [tex]$\displaystyle \frac{5}{12}$[/tex], which is approximately [tex]$0.41667\ldots$[/tex].
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Final Answers:
1. [tex]$$\frac{3}{5}+\frac{1}{3}=\frac{14}{15}\qquad (\approx 0.93333)$$[/tex]
2. The ascending order is: [tex]$$\frac{1}{5},\quad \frac{1}{2},\quad \frac{7}{10},\quad \frac{4}{5},\quad \frac{9}{10}.$$[/tex]
3. [tex]$$\frac{3}{14}=\frac{15}{70}\quad \text{(i.e., }x=15\text{)}.$$[/tex]
4. [tex]$$\frac{3}{4}-\frac{1}{3}=\frac{5}{12}\qquad (\approx 0.41667).$$[/tex]
