Solving for Total Dogs: Analyzing 1,220,500 Dalmatian Spots Using Fractions

Word Problems with Mixed Fractions

A new study has revealed the number of spots that Dalmatians have.

One out of every four dogs has 708 spots.

13 \frac{1}{3} of the dogs have only 660 spots, while 625 of the remaining dogs that participated in the study have 1000 spots.

If the researchers counted 1,220,500 spots in the study, then how many dogs participated in it?

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Step-by-step written solution

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1

Understand the problem

A new study has revealed the number of spots that Dalmatians have.

One out of every four dogs has 708 spots.

13 \frac{1}{3} of the dogs have only 660 spots, while 625 of the remaining dogs that participated in the study have 1000 spots.

If the researchers counted 1,220,500 spots in the study, then how many dogs participated in it?

2

Step-by-step solution

To solve this problem, we need to determine how many Dalmatians participated based on the number of spots each group has and the total spots counted.

Let's denote the total number of dogs as x x .

According to the data provided:

  • One out of every four dogs has 708 spots, which gives us 14x×708 \frac{1}{4}x \times 708 spots for this group.
  • One out of every three dogs has 660 spots, which gives us 13x×660 \frac{1}{3}x \times 660 spots for this group.
  • 625 dogs have 1000 spots each, contributing 625×1000 625 \times 1000 spots.

We need to sum these contributions to get the total spot count of 1,220,500.

The total equation for spots is:

14x×708+13x×660+625×1000=1,220,500 \frac{1}{4}x \times 708 + \frac{1}{3}x \times 660 + 625 \times 1000 = 1,220,500

Let's simplify this equation:

14x×708=177x \frac{1}{4}x \times 708 = 177x 13x×660=220x \frac{1}{3}x \times 660 = 220x

Thus, the equation becomes:

177x+220x+625×1000=1,220,500 177x + 220x + 625 \times 1000 = 1,220,500 397x+625,000=1,220,500 397x + 625,000 = 1,220,500

Subtract 625,000 from both sides:

397x=595,500 397x = 595,500

Now, divide by 397:

x=595,500397=1500 x = \frac{595,500}{397} = 1500

The number of dogs that participated in the study is therefore 1500 1500 .

3

Final Answer

1500

Key Points to Remember

Essential concepts to master this topic
  • Setup: Define variable x for total, express each group as fraction
  • Technique: Calculate 14x×708+13x×660+625,000=1,220,500 \frac{1}{4}x \times 708 + \frac{1}{3}x \times 660 + 625,000 = 1,220,500
  • Check: Verify 177(1500) + 220(1500) + 625,000 = 1,220,500 ✓

Common Mistakes

Avoid these frequent errors
  • Adding fractions incorrectly when setting up equation
    Don't add 14+13=27 \frac{1}{4} + \frac{1}{3} = \frac{2}{7} of total dogs = wrong groupings! This ignores that fractions represent separate, independent groups. Always keep each fractional group separate in your equation setup.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

\( 3x=18 \)

FAQ

Everything you need to know about this question

How do I know what variable to use for the total?

+

Use x to represent the total number of dogs since that's what the question asks for. Then express each group as a fraction of x: 14x \frac{1}{4}x , 13x \frac{1}{3}x , and 625 dogs.

Why don't the fractions 1/4 and 1/3 add up to all the dogs?

+

Great observation! The fractions 14+13=712 \frac{1}{4} + \frac{1}{3} = \frac{7}{12} of total dogs, leaving 512 \frac{5}{12} for the remaining 625 dogs. This is why we need algebra to solve it!

What if I set up the equation wrong from the start?

+

Read carefully! Each group contributes: (number of dogs in group) × (spots per dog). Make sure you multiply the right fractions by the right spot counts: 708, 660, and 1000.

How do I simplify 177x + 220x?

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These are like terms because they both have the variable x. Just add the coefficients: 177x+220x=397x 177x + 220x = 397x . Think of it like 177 apples + 220 apples = 397 apples!

Is there a faster way to solve this?

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The algebraic method shown is actually the most reliable! You could guess and check, but with large numbers like 1,220,500, setting up the equation systematically saves time and prevents errors.

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