Ruler Length from Colored Segments
A ruler is divided into four colored segments that together cover its entire length: - A blue segment that is $\frac{1}{4}$ of the ruler's total length - A red segment that is $\frac{1}{5}$ of the ruler's total length - A yellow segment that is $\frac{1}{6}$ of the ruler's total length - A green segment that is exactly 23 units long
How long is the ruler?
Hints
- The four segments cover the entire ruler with no gaps. Write an equation saying their lengths sum to $L$.
- Express the blue, red, and yellow lengths as $L/4$, $L/5$, and $L/6$, then collect all $L$ terms on one side.
- After combining fractions with LCD $= 60$, you get $\frac{23}{60} L = 23$, which gives $L$ immediately.
Worked Solution
How to Think About It: The four segments partition the ruler with no overlap and no gap, so their lengths must sum to the total. Three of the four segments are given as fractions of the unknown total $L$, and one has a fixed length. Write one equation in $L$ and solve.
Quick Estimate: The three fractional pieces take up $\frac{1}{4} + \frac{1}{5} + \frac{1}{6}$ of $L$. Finding a common denominator: $\frac{15}{60} + \frac{12}{60} + \frac{10}{60} = \frac{37}{60}$. So the green piece is the remaining $1 - \frac{37}{60} = \frac{23}{60}$ of $L$, and that equals 23 units. Immediately: $L = 60$.
Approach: Set up the partition equation and solve for $L$.
Formal Solution:
Let $L$ be the total length of the ruler. The four segments are: $$\frac{L}{4} + \frac{L}{5} + \frac{L}{6} + 23 = L$$
Combine the fractional terms using LCD $= 60$: $$\frac{15L}{60} + \frac{12L}{60} + \frac{10L}{60} + 23 = L$$ $$\frac{37L}{60} + 23 = L$$ $$23 = L - \frac{37L}{60} = \frac{23L}{60}$$ $$L = \frac{23 \times 60}{23} = 60$$
Answer: The ruler is $\boxed{60}$ units long.
Intuition
This is a classic 'find the whole from a part' problem. The strategy is always the same: express all known quantities as fractions of the unknown total, sum them up, set the remainder equal to the known fixed piece, and solve. The fact that the green segment was 23 units and the remaining fraction happened to be $\frac{23}{60}$ is what makes the arithmetic so clean -- $L = 60$ is essentially forced by the choice of fractions $\frac{1}{4}, \frac{1}{5}, \frac{1}{6}$ and the number 23.
In practice, this pattern shows up in finance whenever you have a portfolio where some positions are fixed dollar amounts and others are fixed percentage allocations. Finding the total portfolio size from the constraint that allocations sum to 100% is exactly this calculation. The algebraic reflex of 'write the partition equation and solve for the unknown whole' is a useful one to have automatic.