Question

mathematics questions

1. a group of market women sell at least one of yam, plantain and maize. 12 of them sell maize, 10 sell yam and 14 sell plantain. 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only while 3 sell all the three items. how many women are in the group?

a. 25
b. 19
c. 18
d. 17

1. if \\(\log_{8}10 = x\\), evaluate \\(\log_{8}5\\) to base in terms of \\(x\\).

a. \\(\frac{1}{2}x\\)
b. \\(x - \frac{1}{4}\\)
c. \\(x - \frac{1}{2}\\)
d. \\(x - \frac{1}{3}\\)

1. solve the inequality \\(2 - x > x^2\\)

a. \\(x < -2\\) or \\(x > 1\\)
b. \\(x > 2\\) or \\(x < -1\\)
c. \\(-1 < x < 2\\)
d. \\(-2 < x < 1\\)

1. find the range of values of \\(x\\) which satisfy the inequalities \\(4x - 7 \le 3x\\) and \\(3x - 4 \le 4x\\)

a. \\(-4 \le x \le 7\\)
b. \\(-7 \le x \le 4\\)
c. \\(x \ge -7\\)
d. \\(-7 \le x \le 6\\)

1. determine the value of \\(x\\) for which \\((x^2 - 1) > 0\\)

a. \\(x < -1\\) or \\(x > 1\\)
b. \\(-1 < x < 1\\)

Explanation:

🆕 New Concept Discovered: Three-Set Venn Diagrams
Using overlapping circles to organize and count items in three categories.

Step 1: Identify the given information

Let the three sets of market women be:

• \( M \): women who sell maize
• \( Y \): women who sell yam
• \( P \): women who sell plantain

From the problem, we have the following data:

• Total maize sellers, \( n(M) = 12 \)
• Total yam sellers, \( n(Y) = 10 \)
• Total plantain sellers, \( n(P) = 14 \)
• Sell plantain and maize, \( n(P \cap M) = 5 \)
• Sell yam and maize, \( n(Y \cap M) = 4 \)
• Sell all three items, \( n(Y \cap P \cap M) = 3 \)
• Sell yam and plantain only, \( n(Y \cap P \cap M') = 2 \)

Step 2: Determine the size of each overlapping region

To find the total number of women, we can break the Venn diagram down into 7 disjoint (non-overlapping) regions:

1. All three items:

$$ n(Y \cap P \cap M) = 3 $$

1. Yam and Plantain only:

Given directly as:

$$ n(Y \cap P \text{ only}) = 2 $$

1. Yam and Maize only:

We are given that 4 women sell yam and maize in total. This includes those who sell all three.

$$ n(Y \cap M \text{ only}) = n(Y \cap M) - n(Y \cap P \cap M) = 4 - 3 = 1 $$

1. Plantain and Maize only:

We are given that 5 women sell plantain and maize in total. This includes those who sell all three.

$$ n(P \cap M \text{ only}) = n(P \cap M) - n(Y \cap P \cap M) = 5 - 3 = 2 $$

1. Yam only:

The total number of yam sellers is 10. We subtract the overlapping regions containing yam:

$$ n(Y \text{ only}) = n(Y) - [n(Y \cap P \text{ only}) + n(Y \cap M \text{ only}) + n(Y \cap P \cap M)] $$

$$ n(Y \text{ only}) = 10 - [2 + 1 + 3] = 10 - 6 = 4 $$

1. Maize only:

The total number of maize sellers is 12. We subtract the overlapping regions containing maize:

$$ n(M \text{ only}) = n(M) - [n(Y \cap M \text{ only}) + n(P \cap M \text{ only}) + n(Y \cap P \cap M)] $$

$$ n(M \text{ only}) = 12 - [1 + 2 + 3] = 12 - 6 = 6 $$

1. Plantain only:

The total number of plantain sellers is 14. We subtract the overlapping regions containing plantain:

$$ n(P \text{ only}) = n(P) - [n(Y \cap P \text{ only}) + n(P \cap M \text{ only}) + n(Y \cap P \cap M)] $$

$$ n(P \text{ only}) = 14 - [2 + 2 + 3] = 14 - 7 = 7 $$

Step 3: Sum all disjoint regions to find the total number of women

Now, we add all 7 mutually exclusive regions together:

$$ \text{Total} = n(Y \text{ only}) + n(M \text{ only}) + n(P \text{ only}) + n(Y \cap P \text{ only}) + n(Y \cap M \text{ only}) + n(P \cap M \text{ only}) + n(Y \cap P \cap M) $$

$$ \text{Total} = 4 + 6 + 7 + 2 + 1 + 2 + 3 $$

$$ \text{Total} = 25 $$

Answer:

A. 25