Question

a cat makes at least 1 sound (meow, hiss, or purr) each day. in the last 23 days:

• the cat hissed on 6 days.
• the cat purred on 12 days.
• the cat meowed on 15 days.
• on 2 days, the cat meowed and hissed but did not purr.
• on 2 days, the cat purred and hissed but did not meow.
• on 1 day, the cat made all 3 sounds.

on how many days did the cat meow and purr but not hiss?

Explanation:

Step1: Define variables for each region in the Venn diagram (hiss - H, purr - P, meow - M)

Let \( x \) be the number of days the cat meowed and purred but not hissed. Let \( a \) be days only hissed, \( b \) only purred, \( c \) only meowed, \( d = 2 \) (meow and hiss, no purr), \( e = 2 \) (purr and hiss, no meow), \( f = 1 \) (all three).

Step2: Calculate the number of days only hissed (\( a \))

Total hissed days: \( a + d + e + f = 6 \). Substitute \( d = 2 \), \( e = 2 \), \( f = 1 \):
\( a + 2 + 2 + 1 = 6 \)
\( a = 6 - 5 = 1 \)

Step3: Calculate the number of days only purred (\( b \))

Total purred days: \( b + e + f + x = 12 \). Substitute \( e = 2 \), \( f = 1 \):
\( b + 2 + 1 + x = 12 \)
\( b = 9 - x \)

Step4: Calculate the number of days only meowed (\( c \))

Total meowed days: \( c + d + f + x = 15 \). Substitute \( d = 2 \), \( f = 1 \):
\( c + 2 + 1 + x = 15 \)
\( c = 12 - x \)

Step5: Total days equation (since total days = 23)

Total days: \( a + b + c + d + e + f + x = 23 \). Substitute \( a = 1 \), \( b = 9 - x \), \( c = 12 - x \), \( d = 2 \), \( e = 2 \), \( f = 1 \):
\( 1 + (9 - x) + (12 - x) + 2 + 2 + 1 + x = 23 \)
Simplify: \( 1 + 9 - x + 12 - x + 2 + 2 + 1 + x = 23 \)
\( 27 - x = 23 \)
Solve for \( x \): \( x = 27 - 23 = 4 \)

Answer:

\( \boldsymbol{4} \)