A 30-foot-long ribbon was cut into three pieces. The first piece is one-third as long as the second peace and the third piece is four feet longer than three times the length of the second piece. How long is the longest ribbon piece?
To solve this problem in the most efficient way, we are going to use algebra. If we let the first piece of ribbon be the letter 𝑥, let’s try and work out an expression for the second piece of ribbon. We’re told the first piece is one-third as long as the second piece. This means that the second piece is three times as long as the first piece. Well, three multiplied by 𝑥 is three 𝑥. So the second piece of ribbon is three 𝑥 feet.
We are also told that the third piece of ribbon is four feet longer than three times the length of the second piece. Well, three times the length of the second piece will be three multiplied by three 𝑥. This gives us nine 𝑥. But we’re also told that the third piece was four feet longer than this. Therefore, our expression for the third piece of ribbon is nine 𝑥 plus four.
We know that the first piece is 𝑥 feet long, the second piece is three 𝑥 feet long, and the third piece is nine 𝑥 plus four feet long. We also know that the total length of the ribbon was 30 feet. This means we can write an equation: 𝑥 plus three 𝑥 plus nine 𝑥 plus four equals 30. Simplifying or grouping the 𝑥s, the like terms, gives us an equation 13 𝑥 plus four equals 30.
We can then balance the equation by subtracting four from both sides. This gives us 13 𝑥 equals 26. Finally, dividing both sides of this equation by 13 leaves us 𝑥 equals two. The value of 𝑥 in this case is two feet or two foot long.
Substituting 𝑥 equals two into our three expressions gives us for the first piece of ribbon two feet, for the second piece of ribbon, three multiplied by two, and for the third piece of ribbon, nine multiplied by two plus four. The first piece of ribbon is two foot long, the second piece of ribbon three multiplied by two is six foot long, and the third piece of ribbon — the longest piece — is nine multiplied by two, 18, plus four 22 foot long. Therefore, the longest of the three pieces is 22 foot long.