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There are 60 books that are either hardcover or softcover. The hardcover books each weigh 26 ounces. The softcore books each weigh 12 ounces. The books weigh 1,056 ounces in all.

How many of each type of book are there?

3 answers

Let's denote the number of hardcover books as \( h \) and the number of softcover books as \( s \). According to the problem, we have the following two equations based on the provided information:

  1. The total number of books: \[ h + s = 60 \]

  2. The total weight of the books: \[ 26h + 12s = 1056 \]

Now, we can solve these equations step by step.

First, we can solve the first equation for \( s \): \[ s = 60 - h \]

Next, we substitute this expression for \( s \) into the second equation: \[ 26h + 12(60 - h) = 1056 \]

Now, we distribute the 12: \[ 26h + 720 - 12h = 1056 \]

Combine the \( h \) terms: \[ 14h + 720 = 1056 \]

Now, we isolate \( h \) by first subtracting 720 from both sides: \[ 14h = 336 \]

Then divide both sides by 14: \[ h = 24 \]

Now that we have \( h \), we can find \( s \): \[ s = 60 - h = 60 - 24 = 36 \]

Thus, the number of hardcover books is \( h = 24 \) and the number of softcover books is \( s = 36 \).

Finally, we can verify the solution:

  • Total number of books: \( 24 + 36 = 60 \) (correct)
  • Total weight: \( 26 \times 24 + 12 \times 36 \)

Calculating the total weight: \[ 26 \times 24 = 624 \quad \text{and} \quad 12 \times 36 = 432 \] \[ 624 + 432 = 1056 \quad \text{(correct)} \]

Thus, the final answer is:

  • Number of hardcover books: \(\boxed{24}\)
  • Number of softcover books: \(\boxed{36}\)
are you sure that is right?

Let's confirm the solution step-by-step once again to ensure accuracy.

We start with these equations based on the information given:

  1. Total number of books: \[ h + s = 60 \]
  2. Total weight of the books: \[ 26h + 12s = 1056 \]

Step 1: Solve for \( s \)

From the first equation, we can express \( s \) in terms of \( h \): \[ s = 60 - h \]

Step 2: Substitute \( s \) into the weight equation

Substituting \( s \) into the second equation: \[ 26h + 12(60 - h) = 1056 \]

Step 3: Distribute and combine like terms

Distributing the 12: \[ 26h + 720 - 12h = 1056 \]

Now, combine the \( h \) terms: \[ (26h - 12h) + 720 = 1056 \] \[ 14h + 720 = 1056 \]

Step 4: Isolate \( h \)

Subtract 720 from both sides: \[ 14h = 336 \]

Now divide by 14: \[ h = \frac{336}{14} = 24 \]

Step 5: Find \( s \)

Now, substituting \( h \) back into the equation for \( s \): \[ s = 60 - h = 60 - 24 = 36 \]

Conclusion

So the solution is:

  • Number of hardcover books \( h = 24 \)
  • Number of softcover books \( s = 36 \)

Verification

Now let's verify the total weight:

  • Weight of hardcover books: \[ 24 \times 26 = 624 \text{ ounces} \]
  • Weight of softcover books: \[ 36 \times 12 = 432 \text{ ounces} \]

Adding these together: \[ 624 + 432 = 1056 \text{ ounces} \]

Both checks confirm our results are correct. Therefore, the answer is indeed:

  • Hardcover books: \(\boxed{24}\)
  • Softcover books: \(\boxed{36}\)

If you have any further questions or need clarification, please let me know!