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Answer :
Let the two numbers be [tex]$a$[/tex] and [tex]$b$[/tex], with the quotient given as
[tex]$$
\frac{a}{b} = \frac{35}{24},
$$[/tex]
and the product given as
[tex]$$
ab = \frac{14}{15}.
$$[/tex]
Step 1. Express One Variable in Terms of the Other
From the quotient, we have
[tex]$$
a = \frac{35}{24}\,b.
$$[/tex]
Step 2. Substitute into the Product
Replace [tex]$a$[/tex] in the product equation:
[tex]$$
\left(\frac{35}{24}\,b\right) \cdot b = \frac{14}{15}.
$$[/tex]
This simplifies to
[tex]$$
\frac{35}{24} \, b^2 = \frac{14}{15}.
$$[/tex]
Step 3. Solve for [tex]$b^2$[/tex]
Multiply both sides of the equation by the reciprocal of [tex]$\frac{35}{24}$[/tex]:
[tex]$$
b^2 = \frac{14}{15} \times \frac{24}{35}.
$$[/tex]
Simplify the product:
- Multiply the numerators: [tex]$14 \times 24 = 336$[/tex].
- Multiply the denominators: [tex]$15 \times 35 = 525$[/tex].
Thus,
[tex]$$
b^2 = \frac{336}{525}.
$$[/tex]
Simplify the fraction by dividing numerator and denominator by their greatest common divisor. Notice that both [tex]$336$[/tex] and [tex]$525$[/tex] are divisible by [tex]$21$[/tex]:
[tex]$$
\frac{336 \div 21}{525 \div 21} = \frac{16}{25}.
$$[/tex]
So,
[tex]$$
b^2 = \frac{16}{25}.
$$[/tex]
Taking the positive square root (assuming the numbers are positive),
[tex]$$
b = \frac{4}{5}.
$$[/tex]
Step 4. Find [tex]$a$[/tex]
Now, substitute [tex]$b = \frac{4}{5}$[/tex] back into the expression for [tex]$a$[/tex]:
[tex]$$
a = \frac{35}{24} \times \frac{4}{5}.
$$[/tex]
Multiply the fractions:
[tex]$$
a = \frac{35 \times 4}{24 \times 5} = \frac{140}{120}.
$$[/tex]
Simplify the fraction by dividing numerator and denominator by [tex]$20$[/tex]:
[tex]$$
a = \frac{7}{6}.
$$[/tex]
Final Answer
The two numbers are:
[tex]$$
a = \frac{7}{6} \quad \text{and} \quad b = \frac{4}{5}.
$$[/tex]
These numbers satisfy both the given product and quotient conditions:
[tex]$$
ab = \frac{7}{6} \times \frac{4}{5} = \frac{28}{30} = \frac{14}{15},
$$[/tex]
[tex]$$
\frac{a}{b} = \frac{\frac{7}{6}}{\frac{4}{5}} = \frac{7}{6} \times \frac{5}{4} = \frac{35}{24}.
$$[/tex]
[tex]$$
\frac{a}{b} = \frac{35}{24},
$$[/tex]
and the product given as
[tex]$$
ab = \frac{14}{15}.
$$[/tex]
Step 1. Express One Variable in Terms of the Other
From the quotient, we have
[tex]$$
a = \frac{35}{24}\,b.
$$[/tex]
Step 2. Substitute into the Product
Replace [tex]$a$[/tex] in the product equation:
[tex]$$
\left(\frac{35}{24}\,b\right) \cdot b = \frac{14}{15}.
$$[/tex]
This simplifies to
[tex]$$
\frac{35}{24} \, b^2 = \frac{14}{15}.
$$[/tex]
Step 3. Solve for [tex]$b^2$[/tex]
Multiply both sides of the equation by the reciprocal of [tex]$\frac{35}{24}$[/tex]:
[tex]$$
b^2 = \frac{14}{15} \times \frac{24}{35}.
$$[/tex]
Simplify the product:
- Multiply the numerators: [tex]$14 \times 24 = 336$[/tex].
- Multiply the denominators: [tex]$15 \times 35 = 525$[/tex].
Thus,
[tex]$$
b^2 = \frac{336}{525}.
$$[/tex]
Simplify the fraction by dividing numerator and denominator by their greatest common divisor. Notice that both [tex]$336$[/tex] and [tex]$525$[/tex] are divisible by [tex]$21$[/tex]:
[tex]$$
\frac{336 \div 21}{525 \div 21} = \frac{16}{25}.
$$[/tex]
So,
[tex]$$
b^2 = \frac{16}{25}.
$$[/tex]
Taking the positive square root (assuming the numbers are positive),
[tex]$$
b = \frac{4}{5}.
$$[/tex]
Step 4. Find [tex]$a$[/tex]
Now, substitute [tex]$b = \frac{4}{5}$[/tex] back into the expression for [tex]$a$[/tex]:
[tex]$$
a = \frac{35}{24} \times \frac{4}{5}.
$$[/tex]
Multiply the fractions:
[tex]$$
a = \frac{35 \times 4}{24 \times 5} = \frac{140}{120}.
$$[/tex]
Simplify the fraction by dividing numerator and denominator by [tex]$20$[/tex]:
[tex]$$
a = \frac{7}{6}.
$$[/tex]
Final Answer
The two numbers are:
[tex]$$
a = \frac{7}{6} \quad \text{and} \quad b = \frac{4}{5}.
$$[/tex]
These numbers satisfy both the given product and quotient conditions:
[tex]$$
ab = \frac{7}{6} \times \frac{4}{5} = \frac{28}{30} = \frac{14}{15},
$$[/tex]
[tex]$$
\frac{a}{b} = \frac{\frac{7}{6}}{\frac{4}{5}} = \frac{7}{6} \times \frac{5}{4} = \frac{35}{24}.
$$[/tex]
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