Examples with solutions for Common Denominators: Worded problems

Exercise #1

A Grandmother buys one strawberry doughnut and one chocolate doughnut for her two grandchildren, Jessy and James.

Jessy eats 16 \frac{1}{6} of the strawberry doughnut, while James eats

13 \frac{1}{3} of the chocolate doughnut.

How much of the doughnuts do they eat in total?

Step-by-Step Solution

To determine how much of the doughnuts they eat in total, let's find the sum of the fractions that represent their consumption.

First, consider Jessy's consumption of the strawberry doughnut: 16 \frac{1}{6} .

Next, consider James's consumption of the chocolate doughnut: 13 \frac{1}{3} .

To add these fractions, we need a common denominator. The denominators are 6 and 3. The least common multiple of these is 6.

Convert 13 \frac{1}{3} to an equivalent fraction with a denominator of 6:

13=1×23×2=26 \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}

Now we have the fractions 16 \frac{1}{6} and 26 \frac{2}{6} .

We can add them since they have the same denominator:

16+26=1+26=36 \frac{1}{6} + \frac{2}{6} = \frac{1 + 2}{6} = \frac{3}{6}

Therefore, in total, Jessy and James eat:

36 \frac{3}{6} of the doughnuts.

The correct answer choice is the one that corresponds to 36 \frac{3}{6} , which is Choice 2.

Thus, the solution to this problem is that they eat 36 \frac{3}{6} of the doughnuts in total.

Answer

36 \frac{3}{6}

Exercise #2

A mother buys two pizzas for her husband and son.

The first pizza is divided into 3 equal slices, while the second is divided into 9 equal slices.

The husband eats 1 slice of the first pizza and the son eats 2 slices of the second pizza.

How much do the father and son eat in total?

Step-by-Step Solution

To solve this problem, we need to express the portions eaten by the husband and the son as fractions of their respective pizzas and then add these fractions.

First, let's express the husband's consumption as a fraction. The husband eats 1 slice from the first pizza, which is divided into 3 equal slices. Therefore, the husband eats:

13\frac{1}{3} of the first pizza.

Next, express the son's consumption as a fraction. The son eats 2 slices from the second pizza, which is divided into 9 equal slices. Therefore, the son eats:

29\frac{2}{9} of the second pizza.

Now, to add these fractions, we need a common denominator. The denominators here are 3 and 9. The least common denominator for these is 9. So, we convert 13\frac{1}{3} to have a denominator of 9:

13=3×13×3=39\frac{1}{3} = \frac{3 \times 1}{3 \times 3} = \frac{3}{9}.

Now, add the fractions:

39+29=59\frac{3}{9} + \frac{2}{9} = \frac{5}{9}.

Therefore, the total amount the husband and the son eat in total is 59\frac{5}{9} of the combined pizzas.

Thus, the correct answer is 59\frac{5}{9}.

Answer

59 \frac{5}{9}

Exercise #3

John runs around a 100-meter track.

He runs his first 50 meters at a speed of 38 km/h.

The next 25 meters, he runs at a speed of 36 km/h.

He runs the last 25 meters at a speed of 34 km/h.

How much of the track does he cover at a speed of 34 km/h?

Step-by-Step Solution

To solve this problem, we'll determine the fraction of the track John covers at a speed of 34 km/h.

  • Step 1: Understand that John runs three segments of the track:
    • 50 meters at 38 km/h
    • 25 meters at 36 km/h
    • 25 meters at 34 km/h
  • Step 2: Identify that the question specifically asks about the third segment, which is 25 meters.
  • Step 3: Calculate the fraction of the track these 25 meters represent out of 100 meters: Fraction at 34 km/h=25 meters100 meters=14 \text{Fraction at 34 km/h} = \frac{25 \text{ meters}}{100 \text{ meters}} = \frac{1}{4}
  • Step 4: The calculation shows that John covers 14\frac{1}{4} of the track at a speed of 34 km/h.
  • Step 5: Double-check by verifying the simplification and ensuring the calculation matches the track setup and speed descriptions.

Therefore, the solution to the problem is 14\frac{1}{4}.

Answer

14 \frac{1}{4}

Exercise #4

A chocolate cake is divided into 8 equal slices.

When Jorge goes to try the cake, he sees that only 3 slices are left.

If Jorge eats 14 \frac{1}{4} of the cake, then how much of the chocolate cake is left?

Step-by-Step Solution

To solve this problem, we need to determine how much of the cake is left after Jorge consumes part of it. We know the following:

  • The cake initially has 8 slices, with only 3 slices left, meaning 38\frac{3}{8} of the cake remains.
  • Jorge eats 14\frac{1}{4} of the entire original cake.

We need to find 14\frac{1}{4} in terms of eighths to subtract it from 38\frac{3}{8}:

14=28\frac{1}{4} = \frac{2}{8} since multiplying both the numerator and the denominator by 2 converts it.

Thus, the amount of cake Jorge eats from the original amount available is equivalent to 28\frac{2}{8} of the cake.

Now, we perform the subtraction to find the remaining portion of the cake:

3828=18\frac{3}{8} - \frac{2}{8} = \frac{1}{8}

Therefore, the amount of cake left after Jorge eats his portion is 18\frac{1}{8} of the original cake. Thus, the final solution is:

18 \frac{1}{8} .

Answer

18 \frac{1}{8}