(randRange( 1, 5 )*N) randFromArray([ 5, 8 ]) randRange( 3, D - 1 ) PIECES*(D/N)

{Many|All|Several} of person(1)'s friends wanted to try the candy bars he(1) brought back from his(1) trip, but there were only PIECES candy bars. person(1) decided to cut the candy bars into pieces so that each person could have \frac{N}{D} of a candy bar.

After cutting up the candy bars, how many friends could person(1) share his(1) candy with?

SOLUTION

We can divide the number of candy bars (PIECES) by the amount person(1) gave to each person (\frac{N}{D} of a bar) to find out how many people he(1) could share with.

 \dfrac{\color{ORANGE}{PIECES \text{ candy bars}}} {\color{BLUE}{\dfrac{N}{D} \text{ bar per person}}} = \color{PINK}{\text{ total people}} 

Dividing by a fraction is the same as multiplying by the reciprocal.

The reciprocal of \dfrac{N}{D} \text{ bar per person} is \dfrac{D}{N} \text{ people per bar}.

 \color{ORANGE}{PIECES\text{ candy bars}} \times \color{GREEN}{\dfrac{D}{N} \text{ people per bar}} = \color{PINK}{\text{total people}} 

\color{PINK}{\dfrac{D * PIECES}{N}\text{ people}} = SOLUTION\text{ people}

By cutting up the candy bars, person(1) could share his(1) candy with SOLUTION of his(1) friends.
(randRange( 3, 6 )*N) randFromArray([ 3, 5 ]) randRange( 2, D - 1 ) YARN*(D/N)

person(1) just found beautiful yarn {for randFromArray([5,20]) percent off }at his(1) favorite yarn store. He(1) can make 1 scarf from \frac{N}{D} of a ball of yarn.

If person(1) buys YARN balls of yarn, how many scarves can he(1) make?

SOLUTION

We can divide the balls of yarn (YARN) by the yarn needed per scarf (\frac{N}{D} of a ball) to find out how many scarves person(1) can make.

 \dfrac{\color{ORANGE}{YARN \text{ balls of yarn}}} {\color{BLUE}{\dfrac{N}{D} \text{ ball per scarf}}} = \color{PINK}{\text{ number of scarves}} 

Dividing by a fraction is the same as multiplying by the reciprocal.

The reciprocal of \dfrac{N}{D} \text{ ball per scarf} is \dfrac{D}{N} \text{ scarves per ball}.

 \color{ORANGE}{YARN\text{ balls of yarn}} \times \color{GREEN}{\dfrac{D}{N} \text{ scarves per ball}} = \color{PINK}{\text{ number of scarves}} 

\color{PINK}{\dfrac{D * YARN}{N}\text{ scarves}} = SOLUTION\text{ scarves}

person(1) can make SOLUTION scarves.

(randRange( 1, 3 )*N) randFromArray([ 2, 5 ]) randRange( 2, D - 1 ) PAINT*(D/N) (randRange( 1, 20 )+SOLUTION)

person(1) decided to paint some of the rooms at his(1) ROOM-room inn, person(1)'s Place. He(1) discovered he(1) needed \frac{N}{D} of a can of paint per room.

If person(1) had PAINT cans of paint, how many rooms could he(1) paint?

SOLUTION

We can divide the cans of paint (PAINT) by the paint needed per room (\frac{N}{D} of a can) to find out how many rooms person(1) could paint.

 \dfrac{\color{ORANGE}{PAINT \text{ cans of paint}}} {\color{BLUE}{\dfrac{N}{D} \text{ can per room}}} = \color{PINK}{\text{ rooms}} 

Dividing by a fraction is the same as multiplying by the reciprocal.

The reciprocal of \dfrac{N}{D} \text{ can per room} is \dfrac{D}{N} \text{ rooms per can}.

 \color{ORANGE}{PAINT\text{ cans of paint}} \times \color{GREEN}{\dfrac{D}{N} \text{ rooms per can}} = \color{PINK}{\text{ rooms}} 

\color{PINK}{\dfrac{D * PAINT}{N}\text{ rooms}} = SOLUTION\text{ rooms}

person(1) could paint SOLUTION rooms.

randRange( 8, 15 ) randRange( 2, D - 1 ) randRange( 3, 7 ) randRange( 2, B - 1 ) getGCD( N, A ) A / GCD1 N / GCD1 getGCD( D, B ) B / GCD2 D / GCD2 ((A*D)/(B*N))

As the swim coach at school(1), person(1) selects which athletes will participate in the state-wide swim relay.

The relay team swims \frac{A}{B} of a mile all together, with each team member responsible for swimming \frac{N}{D} of a mile. The team must complete the swim in \frac{3}{randRange(4,5)} of an hour.

How many swimmers does person(1) need on the relay team?

SOLUTION

To find out how many swimmers person(1) needs on the team, divide the total distance (\frac{A}{B} of a mile) by the distance each team member will swim (\frac{N}{D} of a mile).

 \dfrac{\color{ORANGE}{\dfrac{A}{B} \text{ mile}}} {\color{BLUE}{\dfrac{N}{D} \text{ mile per swimmer}}} = \color{PINK}{\text{ number of swimmers}} 

Dividing by a fraction is the same as multiplying by the reciprocal.

The reciprocal of \dfrac{N}{D} \text{ mile per swimmer} is \dfrac{D}{N} \text{ swimmers per mile}.

 \color{ORANGE}{\dfrac{A}{B}\text{ mile}} \times \color{GREEN}{\dfrac{D}{N} \text{ swimmers per mile}} = \color{PINK}{\text{ number of swimmers}} 

 \dfrac{\color{ORANGE}{A} \cdot \color{GREEN}{D}} {\color{ORANGE}{B} \cdot \color{GREEN}{N}} = \color{PINK}{\text{ number of swimmers}} 

Reduce terms with common factors by dividing the A in the numerator and the N in the denominator by GCD1:

 \dfrac{\color{ORANGE}{\cancel{A}^{SIMP_A}} \cdot \color{GREEN}{D}} {\color{ORANGE}{B} \cdot \color{GREEN}{\cancel{N}^{SIMP_N}}} = \color{PINK}{\text{ number of swimmers}} 

Reduce terms with common factors by dividing the D in the numerator and the B in the denominator by GCD2:

 \dfrac{\color{ORANGE}{SIMP_A} \cdot \color{GREEN}{\cancel{D}^{SIMP_D}}} {\color{ORANGE}{\cancel{B}^{SIMP_B}} \cdot \color{GREEN}{SIMP_N}} = \color{PINK}{\text{ number of swimmers}} 

Simplify:

 \dfrac{\color{ORANGE}{SIMP_A} \cdot \color{GREEN}{SIMP_D}} {\color{ORANGE}{SIMP_B} \cdot \color{GREEN}{SIMP_N}} = \color{PINK}{SOLUTION} 

person(1) needs SOLUTION swimmers on his(1) team.

randRange( 6, 30 ) randRange( 2, D - 1 ) randRange( 3, 5 ) randRange( 2, B - 1 ) getGCD( N, A ) A / GCD1 N / GCD1 getGCD( D, B ) B / GCD2 D / GCD2 ((A*D)/(B*N))

person(1) thought it would be nice to include \frac{N}{D} of a pound of chocolate in each of the holiday gift bags he(1) made for his(1) friends and family.

How many holiday gift bags could person(1) make with \frac{A}{B} of a pound of chocolate?

SOLUTION

To find out how many gift bags person(1) could create, divide the total chocolate (\frac{A}{B} of a pound) by the amount he(1) wanted to include in each gift bag (\frac{N}{D} of a pound).

 \dfrac{\color{ORANGE}{\dfrac{A}{B} \text{ pound of chocolate}}} {\color{BLUE}{\dfrac{N}{D} \text{ pound per bag}}} = \color{PINK}{\text{ number of bags}} 

Dividing by a fraction is the same as multiplying by the reciprocal.

The reciprocal of \dfrac{N}{D} \text{ pound per bag} is \dfrac{D}{N} \text{ bags per pound}.

 \color{ORANGE}{\dfrac{A}{B}\text{ pound}} \times \color{GREEN}{\dfrac{D}{N} \text{ bags per pound}} = \color{PINK}{\text{ number of bags}} 

 \dfrac{\color{ORANGE}{A} \cdot \color{GREEN}{D}} {\color{ORANGE}{B} \cdot \color{GREEN}{N}} = \color{PINK}{\text{ number of bags}} 

Reduce terms with common factors by dividing the A in the numerator and the N in the denominator by GCD1:

 \dfrac{\color{ORANGE}{\cancel{A}^{SIMP_A}} \cdot \color{GREEN}{D}} {\color{ORANGE}{B} \cdot \color{GREEN}{\cancel{N}^{SIMP_N}}} = \color{PINK}{\text{ number of bags}} 

Reduce terms with common factors by dividing the D in the numerator and the B in the denominator by GCD2:

 \dfrac{\color{ORANGE}{SIMP_A} \cdot \color{GREEN}{\cancel{D}^{SIMP_D}}} {\color{ORANGE}{\cancel{B}^{SIMP_B}} \cdot \color{GREEN}{SIMP_N}} = \color{PINK}{\text{ number of bags}} 

Simplify:

 \dfrac{\color{ORANGE}{SIMP_A} \cdot \color{GREEN}{SIMP_D}} {\color{ORANGE}{SIMP_B} \cdot \color{GREEN}{SIMP_N}} = \color{PINK}{SOLUTION} 

person(1) could create SOLUTION gift bags.

randRange( 7, 20 ) randRange( 2, D - 1 ) randRange( 3, 6 ) randRange( 2, B - 1 ) getGCD( N, A ) A / GCD1 N / GCD1 getGCD( D, B ) B / GCD2 D / GCD2 ((A*D)/(B*N))

person(1) works out for \frac{A}{B} of an hour every day. To keep his(1) exercise routines interesting, he(1) includes different types of exercises, such as plural(exercise(1)) and plural(exercise(2)), in each workout.

If each type of exercise takes \frac{N}{D} of an hour, how many different types of exercise can person(1) do in each workout?

SOLUTION

To find out how many types of exercise person(1) could do in each workout, divide the total amount of exercise time (\frac{A}{B} of an hour) by the amount of time each exercise type takes (\frac{N}{D} of an hour).

 \dfrac{\color{ORANGE}{\dfrac{A}{B} \text{ hour}}} {\color{BLUE}{\dfrac{N}{D} \text{ hour per exercise}}} = \color{PINK}{\text{ number of exercises}} 

Dividing by a fraction is the same as multiplying by the reciprocal.

The reciprocal of \dfrac{N}{D} \text{ hour per exercise} is \dfrac{D}{N} \text{ exercises per hour}.

 \color{ORANGE}{\dfrac{A}{B}\text{ hour}} \times \color{GREEN}{\dfrac{D}{N} \text{ exercises per hour}} = \color{PINK}{\text{ number of exercises}} 

 \dfrac{\color{ORANGE}{A} \cdot \color{GREEN}{D}} {\color{ORANGE}{B} \cdot \color{GREEN}{N}} = \color{PINK}{\text{ number of exercises}} 

Reduce terms with common factors by dividing the A in the numerator and the N in the denominator by GCD1:

 \dfrac{\color{ORANGE}{\cancel{A}^{SIMP_A}} \cdot \color{GREEN}{D}} {\color{ORANGE}{B} \cdot \color{GREEN}{\cancel{N}^{SIMP_N}}} = \color{PINK}{\text{ number of exercises}} 

Reduce terms with common factors by dividing the D in the numerator and the B in the denominator by GCD2:

 \dfrac{\color{ORANGE}{SIMP_A} \cdot \color{GREEN}{\cancel{D}^{SIMP_D}}} {\color{ORANGE}{\cancel{B}^{SIMP_B}} \cdot \color{GREEN}{SIMP_N}} = \color{PINK}{\text{ number of exercises}} 

Simplify:

 \dfrac{\color{ORANGE}{SIMP_A} \cdot \color{GREEN}{SIMP_D}} {\color{ORANGE}{SIMP_B} \cdot \color{GREEN}{SIMP_N}} = \color{PINK}{SOLUTION} 

person(1) can do SOLUTION different types of exercise per workout.