We are told that Hana can finish a puzzle in

6

hours and Eric can do it in

5

hours. We will find the time

t

that it would take for them to do the job together. Let's start by finding the unit rate of puzzles per hour that can be completed by Hana and Eric, individually.

Hana ’ s Rate : Eric ’ s Rate : \frac{1 puzzle}{6 hours} \frac{1 puzzle}{5 hours}

Note that the product of rate and time gives the fraction of the job done. We can multiply each rate by the time

t

to represent the amount of the job done by each person.

People	Rate ( $puzzle / hour$ )	Fraction of the Job Done
Hana	$\frac{1}{6}$	$\frac{1}{6} t$
Eric	$\frac{1}{5}$	$\frac{1}{5} t$

The sum of the fractions that represent the amount of the job done by each person will be equal to

1 .

\frac{1}{6} t + \frac{1}{5} t = 1

We will solve this equation by multiplying both sides of the equation by the least common denominator (LCD).

\frac{1}{6} t + \frac{1}{5} t = 1

MultEqn

$LHS \cdot 30 = RHS \cdot 30$

30 (\frac{1}{6} t + \frac{1}{5} t) = 30

▼

Solve for

t

Distr

Distribute $30$

30 (\frac{1}{6} t) + 30 (\frac{1}{5} t) = 30

MoveLeftFacToNumOne

$a \cdot \frac{1}{b} = \frac{a}{b}$

\frac{3 0}{6} t + \frac{3 0}{5} t = 30

CalcQuot

Calculate quotient

5 t + 6 t = 30

AddTerms

Add terms

11 t = 30

DivEqn

$LHS / 11 = RHS / 11$

t = \frac{3 0}{1 1}

UseCalc

Use a calculator

t = 2.727272 \dots

RoundDec

Round to $1$ decimal place(s)

t \approx 2.7

It would take them about

2.7

hours to finish the puzzle together, which corresponds to option D.

Exercise