Three pieces of clay, each 3/4 inch long, are joined end to end. How many 3/16-inch pieces can be cut from the result?

Step-by-step solution

1. 1

   Find the total length first. Joining pieces end to end adds their lengths, so three pieces of $\frac34$ inch give

   $3\times\frac{3}{4}=\frac{9}{4}\ \text{inches}=2\tfrac14\ \text{inches}.$

2. 2

   Recognise the question as a division. "How many pieces of size $b$ fit inside a length $a$?" is measurement division: the answer is $a\div b$. Here

   $\frac{9}{4}\div\frac{3}{16}.$

   It is worth estimating before computing: $\frac{3}{16}$ is a bit less than $\frac14$ inch, and about nine quarters fit in $2\tfrac14$ inches, so the answer should be a little more than $9$.

3. 3

   Divide by multiplying by the reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal, because $\frac{3}{16}\times\frac{16}{3}=1$:

   $\frac{9}{4}\div\frac{3}{16}=\frac{9}{4}\times\frac{16}{3}.$

4. 4

   Cancel before multiplying. Cancel the $3$ into the $9$ and the $4$ into the $16$ to keep the numbers small:

   $\frac{\cancel{9}^{\,3}}{\cancel{4}^{\,1}}\times\frac{\cancel{16}^{\,4}}{\cancel{3}^{\,1}}=3\times 4=12.$

5. 5

   Check by multiplying back. $12\times\frac{3}{16}=\frac{36}{16}=\frac{9}{4}$ inches, exactly the total length — so the $12$ pieces use up the clay with nothing left over, and the estimate of "a little more than $9$" is confirmed.