QuestionA rain gauge is 2/5 filled after 3/4 hours of rain. How much of the gauge will be filled after 15 more minutes of rain at the same rate? Explain why the division equation 3/4 ÷ 2/5 does not represent the situation, and write a multiplication and division equation that does.

Studdy Solution

STEP 1

Assumptions
1. The rain gauge is $\frac{2}{5}$ filled after raining for $\frac{3}{4}$ of an hour.
2. We are asked to find out how much more the rain gauge will be filled after an additional 15 minutes of rain.
3. We need to explain why the division equation $\frac{3}{4} \div \frac{2}{5}$ does not represent the situation.
4. We need to write a multiplication equation and a division equation that correctly represent the situation.

STEP 2

First, let's address part a of the problem by explaining why the division equation $\frac{3}{4} \div \frac{2}{5}$ does not represent the situation.
The division equation suggests that we are trying to find out how many times $\frac{2}{5}$ goes into $\frac{3}{4}$ , which is not related to the problem of finding out how much the rain gauge will fill up after additional rain.

STEP 3

Now, let's move on to part b of the problem by writing a multiplication equation that represents the situation.
We know that the rain gauge is $\frac{2}{5}$ filled after $\frac{3}{4}$ of an hour. To find out how much the rain gauge fills in one hour, we can set up the following multiplication equation:
$Rate\, of\, filling = \frac{Amount\, filled}{Time\, taken}$

STEP 4

Plug in the values for the amount filled and the time taken to calculate the rate of filling.
$Rate\, of\, filling = \frac{\frac{2}{5}}{\frac{3}{4}}$

STEP 5

To perform the division, we multiply by the reciprocal of the divisor.
$Rate\, of\, filling = \frac{2}{5} \times \frac{4}{3}$

STEP 6

Multiply the numerators and the denominators.
$Rate\, of\, filling = \frac{2 \times 4}{5 \times 3}$

STEP 7

Simplify the fraction.
$Rate\, of\, filling = \frac{8}{15}$
This means that the rain gauge fills $\frac{8}{15}$ every hour.

STEP 8

Now, let's write a division equation that represents the situation.
We want to find out how much the rain gauge will fill up in 15 minutes, which is $\frac{1}{4}$ of an hour. We can set up the following division equation:
$Amount\, filled\, in\, 15\, minutes = Rate\, of\, filling \div Time\, taken$

STEP 9

Plug in the values for the rate of filling and the time taken (15 minutes is $\frac{1}{4}$ of an hour) to calculate the amount filled in 15 minutes.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \div \frac{1}{4}$

STEP 10

To perform the division, we multiply by the reciprocal of the divisor.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \times 4$

STEP 11

Multiply the numerators and the denominators.
$Amount\, filled\, in\, 15\, minutes = \frac{8 \times 4}{15}$

STEP 12

Simplify the fraction.
$Amount\, filled\, in\, 15\, minutes = \frac{32}{15}$
However, this result is greater than 1, which is not possible for the fraction of the rain gauge filled in 15 minutes. We made a mistake in our calculation. Let's correct this.

STEP 13

Re-evaluate the multiplication.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \times \frac{1}{4}$

STEP 14

Multiply the numerators and the denominators.
$Amount\, filled\, in\, 15\, minutes = \frac{8 \times 1}{15 \times 4}$

STEP 15

Simplify the fraction.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{60}$

STEP 16

Reduce the fraction to its simplest form.
$Amount\, filled\, in\, 15\, minutes = \frac{2}{15}$

STEP 17

Now, we add this amount to the original $\frac{2}{5}$ to find out the total fraction of the rain gauge that will be filled.
$Total\, filled = Original\, amount + Additional\, amount$

STEP 18

Plug in the values for the original amount and the additional amount to calculate the total filled.
$Total\, filled = \frac{2}{5} + \frac{2}{15}$

STEP 19

Find a common denominator for the fractions.
$Total\, filled = \frac{6}{15} + \frac{2}{15}$

STEP 20

Add the fractions.
$Total\, filled = \frac{6 + 2}{15}$

STEP 21

Simplify the fraction.
$Total\, filled = \frac{8}{15}$
After 15 more minutes of rain, $\frac{8}{15}$ of the rain gauge will be filled.

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QuestionA rain gauge is 2/5 filled after 3/4 hours of rain. How much of the gauge will be filled after 15 more minutes of rain at the same rate? Explain why the division equation 3/4 ÷ 2/5 does not represent the situation, and write a multiplication and division equation that does.

Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Plug in the values for the amount filled and the time taken to calculate the rate of filling.Rate of filling=2534Rate\, of\, filling = \frac{\frac{2}{5}}{\frac{3}{4}}Rateoffilling=43​52​​

STEP 5

To perform the division, we multiply by the reciprocal of the divisor.Rate of filling=25×43Rate\, of\, filling = \frac{2}{5} \times \frac{4}{3}Rateoffilling=52​×34​

STEP 6

Multiply the numerators and the denominators.Rate of filling=2×45×3Rate\, of\, filling = \frac{2 \times 4}{5 \times 3}Rateoffilling=5×32×4​

STEP 7

Simplify the fraction.Rate of filling=815Rate\, of\, filling = \frac{8}{15}Rateoffilling=158​This means that the rain gauge fills 815\frac{8}{15}158​ every hour.

STEP 8

STEP 9

Plug in the values for the rate of filling and the time taken (15 minutes is 14\frac{1}{4}41​ of an hour) to calculate the amount filled in 15 minutes.Amount filled in 15 minutes=815÷14Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \div \frac{1}{4}Amountfilledin15minutes=158​÷41​

STEP 10

To perform the division, we multiply by the reciprocal of the divisor.Amount filled in 15 minutes=815×4Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \times 4Amountfilledin15minutes=158​×4

STEP 11

Multiply the numerators and the denominators.Amount filled in 15 minutes=8×415Amount\, filled\, in\, 15\, minutes = \frac{8 \times 4}{15}Amountfilledin15minutes=158×4​

STEP 12

STEP 13

Re-evaluate the multiplication.Amount filled in 15 minutes=815×14Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \times \frac{1}{4}Amountfilledin15minutes=158​×41​

STEP 14

Multiply the numerators and the denominators.Amount filled in 15 minutes=8×115×4Amount\, filled\, in\, 15\, minutes = \frac{8 \times 1}{15 \times 4}Amountfilledin15minutes=15×48×1​

STEP 15

Simplify the fraction.Amount filled in 15 minutes=860Amount\, filled\, in\, 15\, minutes = \frac{8}{60}Amountfilledin15minutes=608​

STEP 16

Reduce the fraction to its simplest form.Amount filled in 15 minutes=215Amount\, filled\, in\, 15\, minutes = \frac{2}{15}Amountfilledin15minutes=152​

STEP 17

Now, we add this amount to the original 25\frac{2}{5}52​ to find out the total fraction of the rain gauge that will be filled.Total filled=Original amount+Additional amountTotal\, filled = Original\, amount + Additional\, amountTotalfilled=Originalamount+Additionalamount

STEP 18

Plug in the values for the original amount and the additional amount to calculate the total filled.Total filled=25+215Total\, filled = \frac{2}{5} + \frac{2}{15}Totalfilled=52​+152​

STEP 19

Find a common denominator for the fractions.Total filled=615+215Total\, filled = \frac{6}{15} + \frac{2}{15}Totalfilled=156​+152​

STEP 20

Add the fractions.Total filled=6+215Total\, filled = \frac{6 + 2}{15}Totalfilled=156+2​

STEP 21

Simplify the fraction.Total filled=815Total\, filled = \frac{8}{15}Totalfilled=158​After 15 more minutes of rain, 815\frac{8}{15}158​ of the rain gauge will be filled.

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Plug in the values for the amount filled and the time taken to calculate the rate of filling.
$Rate\, of\, filling = \frac{\frac{2}{5}}{\frac{3}{4}}$

To perform the division, we multiply by the reciprocal of the divisor.
$Rate\, of\, filling = \frac{2}{5} \times \frac{4}{3}$

Multiply the numerators and the denominators.
$Rate\, of\, filling = \frac{2 \times 4}{5 \times 3}$

Simplify the fraction.
$Rate\, of\, filling = \frac{8}{15}$
This means that the rain gauge fills $\frac{8}{15}$ every hour.

Plug in the values for the rate of filling and the time taken (15 minutes is $\frac{1}{4}$ of an hour) to calculate the amount filled in 15 minutes.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \div \frac{1}{4}$

To perform the division, we multiply by the reciprocal of the divisor.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \times 4$

Multiply the numerators and the denominators.
$Amount\, filled\, in\, 15\, minutes = \frac{8 \times 4}{15}$

Re-evaluate the multiplication.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{15} \times \frac{1}{4}$

Multiply the numerators and the denominators.
$Amount\, filled\, in\, 15\, minutes = \frac{8 \times 1}{15 \times 4}$

Simplify the fraction.
$Amount\, filled\, in\, 15\, minutes = \frac{8}{60}$

Reduce the fraction to its simplest form.
$Amount\, filled\, in\, 15\, minutes = \frac{2}{15}$

Now, we add this amount to the original $\frac{2}{5}$ to find out the total fraction of the rain gauge that will be filled.
$Total\, filled = Original\, amount + Additional\, amount$

Plug in the values for the original amount and the additional amount to calculate the total filled.
$Total\, filled = \frac{2}{5} + \frac{2}{15}$

Find a common denominator for the fractions.
$Total\, filled = \frac{6}{15} + \frac{2}{15}$

Add the fractions.
$Total\, filled = \frac{6 + 2}{15}$

Simplify the fraction.
$Total\, filled = \frac{8}{15}$
After 15 more minutes of rain, $\frac{8}{15}$ of the rain gauge will be filled.