Question

The ratio of Rock Songs to Dance songs on Jonathan's MP3 player is 5:6. If Jonathan has between 100 and 120 Rock and Dance songs, how many Rock songs does he have?

Answer 1

Jonathan has `50` Rock songs.

Explanation:

Let `R` denote the number of Rock songs and `D` the number of Dance songs.

We are given the following information:

• `R` and `D` are whole non-negative integers (since the number of songs must be whole numbers).

• `R:D = 5:6`

• `100 <= R+D <= 120`

Since `R:D = 5:6`, there is some number `n` such that:

`{ (R = 5n), (D = 6n) :}`

Since `5` and `6` have no common factor greater than `1`, then in order for `R` and `D` to be whole numbers, `n` must also be a whole number.

Note that:

`R+D = 5n+6n = 11n`

So we have:

`100 <= 11n <= 120`

Dividing all parts of this inequality by `11` we find:

`100/11 <= n <= 120/11`

Note that:

`9 = 99/11 < 100/11 <= n <= 120/11 < 121/11 = 11`

So the only possible whole value for `n` is `10`, giving:

`{ (R = 5n = 50), (D = 6n = 60) :}`

So Jonathan has `50` Rock songs.

Answer 2

Jonathan has 50 rock songs.

Explanation:

The ratio defines how big something is compared to something else.

We are given the ratio of rock to dance as 5:6.
This means `5R` songs exist for every `6D` songs.

The ratio also gives us a starting point for calculating exact quantities of the items from a given multiple or range of items.
To do this, simply add the components of the ratio:

In this case `5:6` added gives `5 + 6 = 11`

The range of the total number of songs is given as between 100 and 120. Looking for a number in this range that is also a multiple of 11 gives us `110` which is `11 xx 10`

If we use the same multiplier in the given ratio, then Jonathan

has: `5R xx 10 = 50R` (rock) songs.

He also has `6D xx 10 = 60D` (dance) songs

Total number of songs is `110` which satisfies the given range of the number of songs.

Thanks to George for inspiration to complete this answer.