We are given three sub-problems. a) Express $(\frac{13}{15} - \frac{7}{10})$ as a percentage. b) Factorize the expression $ay - y - a + 1$. c) In a community of 9400 people, the number of women exceeds the number of men by 1500. Find the ratio of men to women.

1. Problem Description

We are given three sub-problems.

a) Express $(\frac{13}{15} - \frac{7}{10})$ as a percentage.

b) Factorize the expression $ay - y - a + 1$.

c) In a community of 9400 people, the number of women exceeds the number of men by

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5

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0. Find the ratio of men to women.

2. Solution Steps

a) To express $(\frac{13}{15} - \frac{7}{10})$ as a percentage, first find the difference between the fractions. The least common multiple (LCM) of 15 and 10 is

3

0. So, $\frac{13}{15} = \frac{13 \times 2}{15 \times 2} = \frac{26}{30}$ and $\frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30}$.

Therefore, $\frac{13}{15} - \frac{7}{10} = \frac{26}{30} - \frac{21}{30} = \frac{26-21}{30} = \frac{5}{30} = \frac{1}{6}$.

To express $\frac{1}{6}$ as a percentage, multiply by 100: $\frac{1}{6} \times 100 = \frac{100}{6} = \frac{50}{3} = 16\frac{2}{3}$.

So, the percentage is $16\frac{2}{3}\%$.

b) To factorize the expression $ay - y - a + 1$, we can use factoring by grouping.

$ay - y - a + 1 = y(a-1) - (a-1) = (a-1)(y-1)$.

c) Let $m$ be the number of men and $w$ be the number of women in the community.

We are given that $m + w = 9400$ and $w = m + 1500$.

Substitute the second equation into the first equation:

$m + (m + 1500) = 9400$.

$2m + 1500 = 9400$.

$2m = 9400 - 1500 = 7900$.

$m = \frac{7900}{2} = 3950$.

Now find the number of women: $w = m + 1500 = 3950 + 1500 = 5450$.

The ratio of men to women is $m:w = 3950:5450$.

Simplify the ratio by dividing both sides by their greatest common divisor (GCD).

Both numbers are divisible by

5

0. $\frac{3950}{50} = 79$ and $\frac{5450}{50} = 109$.

Since 79 is a prime number, and 109 is a prime number, the GCD is

5

0. So, the ratio is $79:109$.

3. Final Answer

a) $16\frac{2}{3}\%$

b) $(a-1)(y-1)$

c) $79:109$