I am a seven-letter word but if you remove four of my letters I will have one left. What word am I?

Find the mistake in the below maths equations

A = 2
A(A-1) = 2(A-1)
A2-A = 2A-2
A2-2A = A-2
A(A-2) = A-2
A = 1

Two guards were on duty outside a barracks. One faced up the road to watch for anyone approaching from the North. The other looked down the road to see if anyone approached from the South. Suddenly one of them said to the other, "Why are you smiling?"

How did he know his companion was smiling?

What number when divided in half becomes zero?

What came first, the chicken or the egg?

A police officer was passing by when he saw a guy hiding. He walked up to him and genially asked, "What is your name?"

"Shut up!" the boy responded.

"Where are your manners?" asked the outraged police officer.

"Up that tree," said the boy discourteously, directing to a neighbouring tree.

You're looking for trouble, aren't you?" said the police officer.

"No, trouble is looking for me!" the boy answered solemnly.

Why is the boy behaving like this?

Find the missing number in the image below.

There are 2 people near the river and both of them want to get on the other side. The boat can only take one of them. But they got across.
How?

The below mentioned letters have been placed in a order using a logic.

DEAD, BALL, FEAR, SHOP

Can you find the logic?

John is selling fruits.
He sells grapes at $30, Dates $25, and a Wolfberry $45.
At what price does he sell fig?

PS: Logical Thinking