The lights and breaks of 30 bicycles were tested. 27 bicycles passed the lights and 21 passed the brakes test. The light and breaks both failed on one bicycle. How many bicycles passed both tests?

Explanation

We are given:

• 27 bicycles passed the lights test.
• 21 bicycles passed the brakes test.
• 1 bicycle failed both tests.
• There are 30 bicycles in total.

Let:

• $L$ be the set of bicycles that passed the lights test.
• $B$ be the set of bicycles that passed the brakes test.

We need to find how many bicycles passed both tests, which is the intersection of sets $L$a nd $B$, or $L cap B$.

The formula for the union of two sets is:

$$|L\cup B|=|L|+|B|-|L\cap B|$$

We know:

• $|L cup B| = 30 - 1 = 29$ (since 1 bicycle failed both tests).
• $|L|=27$
• $|B| = 21$

Substitute the values into the formula:

$$29 = 27 + 21 - |L cap B|$$ $$29 = 48 - |L cap B|$$ $$|L cap B| = 48 - 29 = 19$$

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