check your understanding
1. look at the two sentences below.
if something has stripes, then it is a zebra.
if something is a zebra, then it has stripes.
which of the two sentences, if any, is true? how do you know?
2. decide whether the statement is true or false. if false, give a counter - example.
a. if i have 25 cents, then i have 2 dimes and a nickel.
b. if a number is divisible by 4, then that number is divisible by 2.
c. if a figure is a rectangle, then it has 4 sides.
3. consider the statement: if your body temperature is 103°, then you have a fever.
a. identify the condition and the conclusion of the sentence.
b. write the converse of this statement.
c. is the converse true or false? if true, explain why. if false, give a counter - example.
4. if a soccer game in the knock - out round ends in a draw (tie) at the end of the 90 minutes, the game continues into overtime. is this a biconditional statement? explain why or why not.
example 3 use a conjecture to make a prediction
based on the data in the table, how many residents would you expect to vote in the 7th town council election?
look for a pattern by comparing the ratios of the number of voters to the number of residents for each year. then use the pattern to make a conjecture about the number of residents who will vote in the 7th town election.
the ratios are: $\frac{386}{3511}approx0.110$, $\frac{414}{3790}approx0.109$, $\frac{451}{4085}approx0.110$, $\frac{544}{4907}approx0.111$, $\frac{623}{5562}approx0.112$, $\frac{767}{7014}approx0.109$
conjecture: the number of voters each year is about 11% of the residents.
town council elections voter turnout
| year | total residents | voters |
| ---- | ---- | ---- |
| 1 | 3511 | 386 |
| 2 | 3790 | 414 |
| 3 | 4085 | 451 |
| 4 | 4907 | 544 |
| 5 | 5562 | 623 |
| 6 | 7014 | 767 |
| 7 | 7785 |? |

Question

Accepted Answer

### 1.
# Explanation:
## Step1: Recall the definition of a zebra
A zebra is defined as an animal with stripes. So, "If something is a zebra, then it has stripes" is true by the very definition of a zebra. The statement "If something has stripes, then it is a zebra" is false because there are other animals with stripes (e.g., tigers).
# Answer:
The statement "If something is a zebra, then it has stripes" is true. The statement "If something has stripes, then it is a zebra" is false.

### 2.
## a.
# Explanation:
## Step1: Consider coin - value combinations
There are other combinations of coins that can make 25 cents (e.g., 5 nickels, 1 quarter). So the statement "If I have 25 cents, then I have 2 dimes and a nickel" is false. A counter - example is having 1 quarter.
# Answer:
False. Counter - example: 1 quarter.

## b.
# Explanation:
## Step1: Recall divisibility rules
If a number is divisible by 4, it can be written as $n = 4k$ for some integer $k$. Since $4k=2\times(2k)$, the number is also divisible by 2.
# Answer:
True.

## c.
# Explanation:
## Step1: Recall the definition of a rectangle
By definition, a rectangle is a four - sided polygon (quadrilateral). So if a figure is a rectangle, then it has 4 sides.
# Answer:
True.

### 3.
## a.
# Explanation:
## Step1: Identify parts of a conditional statement
In the statement "If your body temperature is 103°, then you have a fever", the condition is "your body temperature is 103°" and the conclusion is "you have a fever".
# Answer:
Condition: Your body temperature is 103°. Conclusion: You have a fever.

## b.
# Explanation:
## Step1: Write the converse
The converse of a conditional statement "If $p$, then $q$" is "If $q$, then $p$". So the converse of "If your body temperature is 103°, then you have a fever" is "If you have a fever, then your body temperature is 103°".
# Answer:
If you have a fever, then your body temperature is 103°.

## c.
# Explanation:
## Step1: Consider fever temperature range
A fever can be defined by different temperature thresholds, and it is possible to have a fever with a body temperature lower than 103° (e.g., 100.4°). So the converse is false. A counter - example is a person with a body temperature of 100.4° who has a fever.
# Answer:
False. Counter - example: A person with a body temperature of 100.4° has a fever.

### 4.
# Explanation:
## Step1: Recall the definition of a biconditional statement
A biconditional statement is of the form " $p$ if and only if $q$", meaning $p
ightarrow q$ and $q
ightarrow p$. In the statement "If a soccer game in the knock - out round ends in a draw (tie) at the end of the 90 minutes, the game continues into overtime", we only have the forward implication ($p
ightarrow q$ where $p$ is "the game ends in a draw at the end of 90 minutes" and $q$ is "the game continues into overtime"). There is no information about the reverse implication (if the game goes into overtime, it must have been a draw at the end of 90 minutes). So it is not a biconditional statement.
# Answer:
No. There is only a one - way implication and no information about the reverse implication.

### 5.
# Explanation:
## Step1: Use the observed ratio
We observed that the ratio of voters to residents is approximately 0.11. For the 7th town council election, with 7786 residents, we can estimate the number of voters by multiplying the number of residents by the ratio.
## Step2: Calculate the estimated number of voters
Let $N$ be the number of voters. $N = 7786\times0.11=856.46\approx856$
# Answer:
Approximately 856 residents are expected to vote.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

1.

Step1: Recall the definition of a zebra

Step1: Consider coin - value combinations

Step1: Recall divisibility rules

Answer:

2.

a.