# Very musical!

I had always wanted to learn to play the guitar. Recently I purchased a guitar, and enrolled myself in a music school near my home.

At the time of enrolling, I made inquiries and learned there were 20 music teachers in the school. The clerk behind the desk informed me there were ten teachers who taught guitar, nine taught violin, and seven taught flute. That left me confused! Now, I know enough arithmetic to figure out that ten plus nine plus seven is 26. But the clerk had said there were 20 teachers in the school. There seemed to be a catch somewhere!

I asked the clerk about this. He smiled at my foolishness. "There are 20 teachers, but each of them might teach more than one instrument," he explained. Well, even that explanation didn't do much good. The clerk was a patient man and was willing to explain further. "Let's suppose there are three friends A, B and C whose fruit preferences are as follows: A eats only mangoes, B eats mangoes and apples, and C eats mangoes, apples, and bananas. So, how many of them eat mangoes?" That was an easy one for me. All the three A, B and C ate mangoes, so I said, "Three." The clerk was happy with the answer. "That's right. Now tell me how many of them eat apples," he asked. B and C ate apples, so I said, "Two." "Right again. Now, tell me how many of them eat bananas." Only C ate bananas and, so, I said, "One." "Correct," the clerk said. "Now, three plus two plus one is six. So, are there six persons or three?" he asked smilingly. "Three," I said, and the clouds seemed to clear; the mystery surrounding the troublesome figures of 20 and 26 was solved.

"So, in our school there are 20 teachers. Ten teachers teach guitar, nine teach violin, and seven teach flute. Four teachers teach both guitar and violin, but none teach both guitar and flute. Can you tell me how many teachers teach both flute and violin?" That was the clerk's parting shot. "If you give me the correct answer, I will offer you a ten per cent reduction in fee," he said. Can you help me figure out how many teachers teach both flute and violin? Please help.

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in