Exercise 1

Determine the fraction that matches the description below:

a. half of a half

b. half of a third

c. a third of a half

d. half of a quarter

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Exercise 2

To prepare a cake, you need:

\frac{1}{3} of a package of 750 g of sugar

\frac{3}{4} of a 1 kg package of flour

\frac{3}{5} of a bar of butter of weighing 200 g

Find the quantities that are needed to prepare the cake in grams.

Exercise 3

A well contains 150 liters of water. If \frac{2}{5} of its original content is consumed, how many liters of water are left in the well?

Exercise 4

A piece of cloth measuring 48 m in length is cut to \frac{3}{4} of its original length. What is the length of the piece that has been removed?

Exercise 5

A family consumed the following amounts of liquids during their Saturday picnic:

Two bottles of water each measuring a liter and a half

4 bottles of juice each measuring \frac{1}{3} of a liter

5 bottles of lemonade each measuring \frac{1}{4} of a liter

How many liters of liquid did the family drink during their picnic? Express the result using a mixed number.

Exercise 6

How many thirds of a liter are there in 4 liters?

Exercise 7

A cable measuring 72 m is cut into two pieces. One piece of the cable is \frac{5}{12} of the original length. What is the length of each piece of cable?

Exercise 8

A box contains 60 chocolates. Eva ate \frac{1}{4} of the chocolates and Ana ate \frac{1}{3} of the chocolates.

a. How many chocolates did Eva eat?

b. How many chocolates did Ana eat?

c. What fraction of the 60 chocolates have been eaten so far?

Exercise 9

Adriana has walked 600 m on her way to school. This distance is \frac{3}{4} of the total distance from her house to the school. What is the total distance from her house to the school?

Exercise 10

Two cars A and B are making the same journey of 572 km. At a particular moment, Car A has traveled \frac{5}{11} of the trip, while car B has traveled \frac{6}{13} of it.

a. If the cars continue at the same speed, which of the two will arrive at the destination first?

b. How many kilometers had each car driven at that particular moment?

Exercise 11

After the ballots have been counted in a city's local election, it was determined that \frac{3}{11} of the votes went to Party A, \frac{3}{10} went to Party B, \frac{5}{14} went to Party C and the remainder went to Party D. The total number of votes was 15,400.

Calculate:

a. The number of votes obtained by each party.

b. The number of registered voters who did not cast a ballot if the total amount of voters that did was \frac{5}{8} of the total amount of voters who are registered.

Exercise 12

Helen went to the women's boutique starting out with 180 dollars. She spent \frac{3}{5} of that amount on new clothes. How much money does Helen have left?

Exercise 13

A few years ago Peter was 24, which represents \frac{2}{3} of the amount of his current age. How old is Peter now?

Exercise 14

A father has divided up 1,800 dollars among his 3 sons. The eldest son received \frac{4}{9} of the whole amount, while the middle son received \frac{1}{3} and the youngest son received the remainder.

a. How much money did each sibling receive?

b. What fraction of the money did the youngest son receive?

Exercise 15

The budget for a residential complex is distributed as follows: \frac{2}{5}th for electricity, \frac{1}{4}th for water, \frac{1}{12}th for garbage collection, \frac{1}{8}th for building maintenance and the remainder is reserved for cleaning the complex.

a. What fraction of the budget is used for cleaning?

b. Order the expenses from least to greatest.

Exercise 16

Alicia had 300 dollars to go shopping with. Thursday she spent \frac{2}{5} of that amount and Saturday she spent \frac{3}{4} of what she had left.

a. How much money did she spend each day?

b. How much money does she have left at the end?

Exercise 1 Solution

Determine the fraction that matches the description below:

 a. half a half

(\frac{1}{2})(\frac{1}{2})=\frac{1}{4}

 b. half of a third

(\frac{1}{2})(\frac{1}{3})=\frac{1}{6}

 c. a third of a half

(\frac{1}{3})(\frac{1}{2})=\frac{1}{6}

 d. half of a quarter

(\frac{1}{2})(\frac{1}{4})=\frac{1}{8}

Exercise 2 Solution

To prepare a cake, you need:

\frac{1}{3} of a package of 750 g of sugar

\frac{3}{4} of a 1 kg package of flour

\frac{3}{5} of a bar of butter of 200 g

Find the quantities that are needed to prepare the cake in grams.

\frac{1}{3} of 750 g is \frac{1}{3}(750)=250 g of sugar

\frac{3}{4} of 1000 g is \frac{3}{4}(1000)=750 g of flour

\frac{3}{5} of 200 g is \frac{3}{5}(200)=120 g of butter

Exercise 3 Solution

A well contains 150 liters of water. If \frac{2}{5} of its original content is consumed, how many liters of water are left in the well?

\frac{5}{5}-\frac{2}{5}=\frac{3}{5} is the fractional amount left in the well.

\frac{3}{5}(150)=90 liters is the amount of water left in the well.

Exercise 4 Solution

A piece of cloth measuring 48 m in length is cut to \frac{3}{4} of its original length. What is the length of the piece that has been removed?

\frac{4}{4}-\frac{3}{4}=\frac{1}{4} is the fractional amount of the piece that was removed.

\frac{1}{4}(48)=12 m is the length of the piece that was removed.

Exercise 5 Solution

A family consumed the following amounts of liquids during their Saturday picnic:

Two bottles of water each measuring a liter and a half

4 bottles of juice each measuring \frac{1}{3} of a liter

5 bottles of lemonade each measuring \frac{1}{4} of a liter

How many liters of liquid did the family drink during their picnic? Express the result using a mixed number.

amount of water drank: 2(1\frac{1}{2})=3 liters

amount of juice drank: 4(\frac{1}{3})=\frac{4}{3} liters

amount of lemonade drank: 5(\frac{1}{4})=\frac{5}{4} liters

total amount of liquids drank: 2+\frac{4}{3}+\frac{5}{4}=\frac{24}{12}+\frac{16}{12}+\frac{15}{12}=\frac{24+16+15}{12}=\frac{55}{12} liters

\frac{55}{12}=4 \frac{7}{12} liters

Exercise 6 Solution

How many thirds of a liter are there in 4 liters?

There are three \frac{1}{3} liters in 1=\frac{3}{3} liter.

There are 4\times 3=12 thirds \frac{1}{3}'s of a liter in 4 liters.

Exercise 7 Solution

A cable measuring 72 m is cut into two pieces. One piece of the cable is \frac{5}{12} of the original length. What is the length of each piece of cable?

The fractional length of the second piece of cable: \frac{12}{12}-\frac{5}{12}=\frac{7}{12}.

The 2 pieces are \frac{5}{12}th and \frac{7}{12}th of the original length.

The length of the first piece: \frac{5}{12}(72)=30 m.

     The length of the second piece: \frac{7}{12}(72)=42 m.

Exercise 8 Solution

A box contains 60 chocolates. Eva ate \frac{1}{4} of the chocolates and Ana ate \frac{1}{3} of the chocolates.

a. How many chocolates did Eva eat?

Eva ate \frac{1}{4}(60)=15 chocolates.

b. How many did Ana eat?.

Ana ate \frac{1}{3}(60)=20 chocolates

c. What fraction of the total amount of chocolates has been eaten between them?

\frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}

Exercise 9 Solution

Adriana has walked 600 m on her way to school. This distance is \frac{3}{4}th of the total distance from her house to the school. What is the total distance from her house to the school?

\frac{3}{4} of the total distance plus \frac{1}{4} of the total distance will be the actual total distance.

\frac{1}{4} is \frac{1}{3} of \frac{3}{4}

Find \frac{1}{3} of the distance 600 m.

\frac{1}{3} of 600 m is 200 m

The total distance is 600+200=800 m.

Exercise 10 Solution

Two cars A and B make the same journey of 572 km. At a particular moment, Car A has traveled \frac{5}{11} of the trip, while Car B has traveled \frac{6}{13} of it.

a. If the cars continue at the same speed, which of the two will arrive at the destination first?

Car A has traveled \frac{5}{11} of the total trip and car B has traveled \frac{6}{13} of the total trip.

Car A: \frac{13}{13}(\frac{5}{11})=\frac{65}{143} of the trip

Car B: \frac{11}{11}(\frac{6}{13})=\frac{66}{143} of the trip

\frac{65}{143}<\frac{66}{143} so Car B will make it to the end of the journey just a hair faster than Car A.

b. How many kilometers have been traveled at this particular moment by each car?

Car A: \frac{65}{143}(572)=65(4)=260 km

Car B: \frac{66}{143}(572)=66(4)=264 km

Exercise 11 Solution

After the ballots have been counted in a city's local election, it was determined that \frac{3}{11}th of the votes went to Party A, \frac{3}{10}th went to Party B, \frac{5}{14}th went to Party C and the remainder went to Party D. The total number of votes was 15,400.

Calculate:

a. The number of votes obtained by each party.

Party A has \frac{3}{11}(15,400)=4200 votes

Party B has \frac{3}{10}(15,400)=4620 votes

Party C has \frac{5}{14}(15,400)=5500 votes

Party D has 15,400-4200-4620-5500=1080 votes

b. The number of registered voters who did not cast a ballot if the number of voters who did was \frac{5}{8} of the total amount of voters who are registered.

\frac{5}{8} of the total amount of registered voters is 15,400 voters,

so \frac{1}{5} of the total amount of people who voted, 15,400, will be \frac{1}{8} of the total amount of registered voters.

\frac{1}{5}(15,400)=3080 is \frac{1}{8} of the registered voters,

so 8(3080)=24,640 is the total amount of registered voters.

Exercise 12 Solution

Helen went to the women's boutique starting out with 180 dollars. She spent \frac{3}{5} of that amount. How much money does she have left?

\frac{5}{5}-\frac{3}{5}=\frac{2}{5}

\frac{2}{5}(180)=72 dollars

Exercise 13 Solution

A few years ago Peter was 24, which represents \frac{2}{3} of his current age. How old is Peter now?

\frac{2}{3} of Peter's current age plus \frac{1}{3} of Peter's current age will be Peter's actual current age.

\frac{1}{3} is \frac{1}{2} of \frac{2}{3}

\frac{1}{2} of 24 is 12

present age is 24+12=36

Exercise 14 Solution

A father has divided up 1,800 dollars among his 3 sons. The eldest son received \frac{4}{9} of the whole amount, while the middle son received \frac{1}{3} and the youngest son received the remainder.

a. How much money did each sibling receive?

The eldest son received \frac{4}{9}(1800)=800 dollars.

The middle son received \frac{1}{3}(1800)=600 dollars.

The youngest son received (\frac{9}{9}-\frac{4}{9}-\frac{3}{9})(1800)=\frac{2}{9}(1800)=400 dollars.

b. What fraction of the money was the amount given to the youngest son?

The youngest son received \frac{2}{9} of the original amount.

Exercise 15 Solution

The budget for a residential complex is distributed as follows: \frac{2}{5}th for electricity, \frac{1}{4}th for water, \frac{1}{12}th for garbage collection, \frac{1}{8}th for building maintenance and the remainder is reserved for cleaning the complex.

a. What fraction of the budget is used for cleaning?

Budget without cleaning expenses: \frac{2}{5}+\frac{1}{4}+\frac{1}{12}+\frac{1}{8}=

(\frac{24}{24})(\frac{2}{5})+(\frac{30}{30})(\frac{1}{4})+(\frac{1}{12})(\frac{10}{10})+(\frac{1}{8})(\frac{15}{15})=

\frac{48}{120}+\frac{30}{120}+\frac{10}{120}+\frac{15}{120}=\frac{103}{120}

Cleaning expenses: \frac{120}{120}-\frac{103}{120}=\frac{17}{120}

b. Order the expenses from least to greatest.

\frac{10}{120}<\frac{15}{120}<\frac{17}{120}<\frac{30}{120}<\frac{48}{120}

garbage < maintenance < cleaning < water < electricity

Exercise 16 Solution

Alicia had 300 dollars to go shopping. Thursday she spent \frac{2}{5}th of that amount and Saturday she spent \frac{3}{4}th of what she had left.

a. How much money did she spend each day?

Thursday Alicia spent \frac{2}{5}(300)=120 dollars.

She has 300-120=180 dollars left for more shopping.

Saturday Alicia spent \frac{3}{4}(180)=135 dollars.

b. How much does she have left at the end?

She has 180-135=45 dollars left.

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