Question

8.7.ip-18
estimate the product. then, find the product.
\\(\left(\dfrac{1}{9} + 2\dfrac{2}{3}\
ight) \times \left(1\dfrac{1}{4} - \dfrac{1}{2}\
ight)\\)
b. the estimated product is 3.
c. the estimated product is 6.
d. the estimated product is 1.
\\(\left(\dfrac{1}{9} + 2\dfrac{2}{3}\
ight) \times \left(1\dfrac{1}{4} - \dfrac{1}{2}\
ight) = \square\\)
(type a whole number, fraction, or mixed number.)
enter your answer in the answer box and then click check answer.
all parts showing

Explanation:

Step1: Simplify the first parenthesis

Convert the mixed number to an improper fraction: \(2\frac{2}{3}=\frac{8}{3}\). Then add \(\frac{1}{9}\) and \(\frac{8}{3}\). Find a common denominator, which is 9. \(\frac{8}{3}=\frac{24}{9}\). So \(\frac{1}{9}+\frac{24}{9}=\frac{25}{9}\).

Step2: Simplify the second parenthesis

Convert the mixed number to an improper fraction: \(1\frac{1}{4}=\frac{5}{4}\). Then subtract \(\frac{1}{2}\) from \(\frac{5}{4}\). Find a common denominator, which is 4. \(\frac{1}{2}=\frac{2}{4}\). So \(\frac{5}{4}-\frac{2}{4}=\frac{3}{4}\).

Step3: Multiply the two results

Multiply \(\frac{25}{9}\) and \(\frac{3}{4}\). \(\frac{25}{9}\times\frac{3}{4}=\frac{25\times3}{9\times4}=\frac{75}{36}\). Simplify this fraction by dividing numerator and denominator by 3: \(\frac{25}{12}=2\frac{1}{12}\)? Wait, no, wait. Wait, 75 divided by 3 is 25, 36 divided by 3 is 12. But wait, maybe I made a mistake. Wait, let's check again. Wait, first parenthesis: \(\frac{1}{9}+2\frac{2}{3}\). \(2\frac{2}{3}\) is \(2 + \frac{2}{3}\), so \(\frac{1}{9}+\frac{2}{3}=\frac{1}{9}+\frac{6}{9}=\frac{7}{9}\), then add 2: \(2+\frac{7}{9}=2\frac{7}{9}\)? Wait, oh no! I made a mistake earlier. Let's correct that.

Step1 (corrected): Simplify the first parenthesis

\(2\frac{2}{3}\) is \(2+\frac{2}{3}\). So \(\frac{1}{9}+2\frac{2}{3}=2 + (\frac{1}{9}+\frac{2}{3})\). \(\frac{2}{3}=\frac{6}{9}\), so \(\frac{1}{9}+\frac{6}{9}=\frac{7}{9}\). So the first parenthesis is \(2\frac{7}{9}=\frac{2\times9 + 7}{9}=\frac{25}{9}\). Wait, that part was correct. Then the second parenthesis: \(1\frac{1}{4}-\frac{1}{2}\). \(1\frac{1}{4}\) is \(\frac{5}{4}\), \(\frac{1}{2}\) is \(\frac{2}{4}\), so \(\frac{5}{4}-\frac{2}{4}=\frac{3}{4}\). Then multiply \(\frac{25}{9}\times\frac{3}{4}\). The 3 and 9 can be simplified: 3 divides into 3 once and into 9 three times. So \(\frac{25}{3}\times\frac{1}{4}=\frac{25}{12}\). \(\frac{25}{12}\) is \(2\frac{1}{12}\)? Wait, but the problem says "Type a whole number, fraction, or mixed number." Wait, but maybe I made a mistake in the first parenthesis. Wait, let's re - do the first parenthesis:

\(\frac{1}{9}+2\frac{2}{3}\). Let's convert \(2\frac{2}{3}\) to ninths: \(2\frac{2}{3}=2\frac{6}{9}\). Then \(\frac{1}{9}+2\frac{6}{9}=2\frac{7}{9}=\frac{25}{9}\). That's correct. Second parenthesis: \(1\frac{1}{4}-\frac{1}{2}\). \(1\frac{1}{4}=\frac{5}{4}\), \(\frac{1}{2}=\frac{2}{4}\), so \(\frac{5}{4}-\frac{2}{4}=\frac{3}{4}\). Then \(\frac{25}{9}\times\frac{3}{4}\). The 3 in the numerator and 9 in the denominator: GCD(3,9)=3. So divide 3 by 3 to get 1, divide 9 by 3 to get 3. So now we have \(\frac{25}{3}\times\frac{1}{4}=\frac{25}{12}\). Wait, but \(\frac{25}{12}\) is \(2\frac{1}{12}\), but that's a mixed number. But maybe I made a mistake in the problem - solving process. Wait, let's check the estimation first. The first parenthesis: \(\frac{1}{9}\) is about 0, \(2\frac{2}{3}\) is about 3. The second parenthesis: \(1\frac{1}{4}\) is about 1, \(\frac{1}{2}\) is 0.5, so 1 - 0.5 = 0.5. Then 3×0.5 = 1.5, which is close to 2 or 1? Wait, the options for estimation: B is 3, C is 6, D is 1. Wait, maybe my estimation was wrong. But let's get back to the exact product.

Wait, \(\frac{25}{9}\times\frac{3}{4}\). Let's compute 25×3 = 75, 9×4 = 36. 75÷3 = 25, 36÷3 = 12. So \(\frac{25}{12}\). Wait, but \(\frac{25}{12}\) is \(2\frac{1}{12}\), but that's not a whole number. Wait, maybe I made a mistake in the first parenthesis. Wait, \(\frac{1}{9}+2\frac{2}{3}\). Let's add them as mixed numbers. \(2\frac{2}{3}+\frac{1}{9}\). The fraction part: \(\frac{2}{3}+\frac{1}{9}=\frac{6}{9}+\frac{1}{9}=\frac{7}{9}\). So it's \(2\frac{7}{9}\), which is \(\frac{25}{9}\). Correct. Second parenthesis: \(1\frac{1}{4}-\frac{1}{2}\). \(1\frac{1}{4}-\frac{2}{4}=1\frac{-1}{4}\)? No, wait, no. \(1\frac{1}{4}\) is 1 + \(\frac{1}{4}\), \(\frac{1}{2}\) is \(\frac{2}{4}\). So 1 + \(\frac{1}{4}\)-\(\frac{2}{4}\)=1 - \(\frac{1}{4}\)=\(\frac{3}{4}\). Oh! Wait a minute! I made a mistake here. \(1\frac{1}{4}-\frac{1}{2}\) is 1 + \(\frac{1}{4}\)-\(\frac{2}{4}\)=1 - \(\frac{1}{4}\)=\(\frac{3}{4}\)? No, 1 is \(\frac{4}{4}\), so \(\frac{4}{4}+\frac{1}{4}-\frac{2}{4}=\frac{3}{4}\). Wait, that's correct. So then \(\frac{25}{9}\times\frac{3}{4}\). Let's do the multiplication again. 25 and 4 are numerator and denominator, 3 and 9 are numerator and denominator. 3 and 9 have a common factor of 3. So 3÷3 = 1, 9÷3 = 3. So now we have \(\frac{25}{3}\times\frac{1}{4}=\frac{25}{12}\). Wait, but \(\frac{25}{12}\) is approximately 2.083. But the problem says "Type a whole number, fraction, or mixed number." So \(\frac{25}{12}\) is \(2\frac{1}{12}\), but that's a mixed number. Wait, maybe I made a mistake in the first parenthesis. Wait, let's check the first parenthesis again. \(\frac{1}{9}+2\frac{2}{3}\). \(2\frac{2}{3}\) is \(\frac{8}{3}\). \(\frac{1}{9}+\frac{8}{3}=\frac{1 + 24}{9}=\frac{25}{9}\). Correct. Second parenthesis: \(1\frac{1}{4}-\frac{1}{2}=\frac{5}{4}-\frac{2}{4}=\frac{3}{4}\). Correct. Then \(\frac{25}{9}\times\frac{3}{4}=\frac{25\times3}{9\times4}=\frac{75}{36}=\frac{25}{12}=2\frac{1}{12}\). Wait, but the problem says "Type a whole number, fraction, or mixed number." So that's the answer. But wait, maybe I made a mistake in the estimation. The first parenthesis: \(\frac{1}{9}\) is about 0, \(2\frac{2}{3}\) is about 3. The second parenthesis: \(1\frac{1}{4}\) is about 1, \(\frac{1}{2}\) is about 0.5, so 1 - 0.5 = 0.5. Then 3×0.5 = 1.5, which is close to 2, but the options for estimation: B is 3, C is 6, D is 1. So the estimated product is about 2, but the options given are B (3), C (6), D (1). Wait, maybe my estimation is wrong. Let's see: \(\frac{1}{9}\) is small, so \(2\frac{2}{3}+\frac{1}{9}\) is about \(2\frac{2}{3}\approx3\). \(1\frac{1}{4}-\frac{1}{2}\) is about \(1 - 0.5 = 0.5\). So 3×0.5 = 1.5, which is close to 2, but the options have D as 1, B as 3. Maybe the estimation is done by rounding to the nearest whole number. \(2\frac{2}{3}\) is close to 3, \(\frac{1}{9}\) is close to 0, so first parenthesis is about 3. \(1\frac{1}{4}\) is close to 1, \(\frac{1}{2}\) is 0.5, so second parenthesis is about 1 - 0.5 = 0.5, which is close to 1? No, 0.5 is close to 1? No, 0.5 is half. Wait, maybe the estimation is done by rounding the mixed numbers to the nearest whole number. \(2\frac{2}{3}\) is close to 3, \(1\frac{1}{4}\) is close to 1, \(\frac{1}{9}\) is close to 0, \(\frac{1}{2}\) is close to 0. So (0 + 3)×(1 - 0)=3×1 = 3. So the estimated product is 3 (option B). Then the exact product is \(\frac{25}{12}=2\frac{1}{12}\)? Wait, no, wait, I think I made a mistake in the second parenthesis. Wait, \(1\frac{1}{4}-\frac{1}{2}\): \(1\frac{1}{4}\) is 1.25, \(\frac{1}{2}\) is 0.5, so 1.25 - 0.5 = 0.75, which is \(\frac{3}{4}\). Correct. First parenthesis: \(\frac{1}{9}+2\frac{2}{3}\): \(2\frac{2}{3}\) is 2.666..., \(\frac{1}{9}\) is 0.111..., so sum is 2.777..., which is \(\frac{25}{9}\approx2.777\). Then 2.777...×0.75≈2.083, which is \(\frac{25}{12}\approx2.083\). But the problem says "Type a whole number, fraction, or mixed number." So \(\frac{25}{12}\) or \(2\frac{1}{12}\). Wait, but maybe I made a mistake in the calculation. Let's check again:

First parenthesis:
\(\frac{1}{9}+2\frac{2}{3}=\frac{1}{9}+\frac{8}{3}=\frac{1 + 24}{9}=\frac{25}{9}\) (correct)

Second parenthesis:
\(1\frac{1}{4}-\frac{1}{2}=\frac{5}{4}-\frac{2}{4}=\frac{3}{4}\) (correct)

Multiplication:
\(\frac{25}{9}\times\frac{3}{4}=\frac{25\times3}{9\times4}=\frac{75}{36}=\frac{25}{12}=2\frac{1}{12}\) (correct)

So the exact product is \(\frac{25}{12}\) or \(2\frac{1}{12}\). But wait, the problem says "Type a whole number, fraction, or mixed number." So that's the answer.

Answer:

The estimated product is 3 (option B). The exact product is \(\frac{25}{12}\) (or \(2\frac{1}{12}\)). Wait, but let's check the problem again. Wait, the problem says "Estimate the product. Then, find the product." For the estimation, as we saw, (0 + 3)×(1 - 0.5)=3×0.5 = 1.5, but if we round the first parenthesis to 3 (since \(2\frac{2}{3}\) is close to 3) and the second parenthesis to 1 (since \(1\frac{1}{4}-\frac{1}{2}=0.75\) is close to 1), then 3×1 = 3. So the estimated product is 3 (option B). Then the exact product is \(\frac{25}{12}\) (or \(2\frac{1}{12}\)). But let's confirm the multiplication once more:

\(\frac{25}{9}\times\frac{3}{4}\):

25×3 = 75

9×4 = 36

75÷3 = 25

36÷3 = 12

So \(\frac{25}{12}\) is the simplified fraction. Yes.

So the final answer for the product is \(\frac{25}{12}\) (or \(2\frac{1}{12}\)). But let's check if we made a mistake in the first parenthesis. Wait, \(\frac{1}{9}+2\frac{2}{3}\):

\(2\frac{2}{3}=\frac{8}{3}=\frac{24}{9}\)

\(\frac{1}{9}+\frac{24}{9}=\frac{25}{9}\). Correct.

Second parenthesis:

\(1\frac{1}{4}=\frac{5}{4}\)

\(\frac{5}{4}-\frac{1}{2}=\frac{5}{4}-\frac{2}{4}=\frac{3}{4}\). Correct.

Multiplication:

\(\frac{25}{9}\times\frac{3}{4}=\frac{25\times3}{9\times4}=\frac{75}{36}=\frac{25}{12}\). Correct.

So the product is \(\frac{25}{12}\) (or \(2\frac{1}{12}\)).