Question

Each of the dinners served at a banquet was either chicken or beef or fish. The ratio of the number of chicken dinners to the number of beef dinners to the number of fish dinners served at the banquet was 7:5:2, respectively. If there were more than 5 fish dinners served at the banquet,

what was the total number of dinners served at the banquet?
(1) The total number of beef dinners and fish dinners served at the banquet was less than 30.
(2) The number of chicken dinners served at the banquet was less than 25.

Answer 1

42 dinners

Explanation:

The ratio chicken:beef:fish is 7k:5k:2k

More than 5 fish dinners were served. The smallest whole number greater than 5 is 6. At least 6 fish dinners were served.

The ratio of the dinners is 7k:5k:2k

2k >= 6

k >= 3

Using k = 3, we get:

21:15:6, and 21 + 15 + 6 = 42 total meals

Then beef + fish = 21 + 6 = 27 which is less than 30.

15 is less than 25

Both conditions are met.

Using k = 4, we get:

28:20:8, and 28 + 20 + 8 = 56 total meals

Then beef + fish = 28 + 8 = 36 which is greater than 30.

The first condition is not met, so this cannot be the answer.

Answer: 42 dinners

Answer 2

Total dinners served at the banquet = 42

Explanation:

chicken:beef:fish = 7:5:2

Chicken =
`(7)/(2) Fish` =
`(7)/(5) Beef`

Beef =
`(5)/(2) Fish` =
`(5)/(7) Chicken`

Fish =
`(2)/(7) Chicken` =
`(2)/(5) Beef`

Fish > 5...........(i)

Beef + Fish < 30

`(5)/(2) Fish` + Fish < 30

Fish < 8.57........(ii)

Beef +
`(2)/(5) Beef` < 30

Beef < 21.43.........(iii)

`(5)/(7) Chicken` +
`(2)/(7) Chicken` < 30

Chicken < 30.........(iv)

Chicken < 25........(v)

`(7)/(2) Fish` < 25

Fish < 7.143........(vi)

`(7)/(5) Beef` < 25

Beef < 17.857........(vii)

From (i), (ii) and (vi)

Fish > 5; Fish < 8.57; Fish <7.143;

Therefore: 5 < Fish < 7.143

So fish is either 6 or 7

From (iii) and (vii)

Beef < 21.43; Beef < 17.857;

Therefore Beef < 17.857

If Fish = 6, Beef = 15;

If Fish = 7, Beef = 17.5

From (iv) and (v)

Chicken < 30; Chicken <25;

Therefore Chicken < 25

If Fish = 6, Chicken = 21;

If Fish = 7, Beef = 24.5

Since meals must be whole numbers, Fish = 6, Beef = 15, Chicken = 21

Total = chicken + beef + fish = 21 + 15 + 6 = 42