## Envision Math Accelerated Grade 7 Volume Chapter 1 Integers and Rational Numbers

**Question. Find the surfboard width between two surfboard models. Lindy’s board is \(23 \frac{1}{3}\) inches wide. Her board is between the 92 and 102 models.**

**Given:
**

To find the surfboard width between two surfboard models.

Lindy’s board is \(23 \frac{1}{3}\) inches wide. Her board is between the 92 and 102 models.

It is like this because

\(23 \frac{1}{3}\) < 24

And

\(23 \frac{1}{4}\) < \(24 \frac{1}{3}\)

The first one is obvious regarding to the second one \(\frac{1}{4}\)=0.25 < \(\frac{1}{3}\)=0.3333.

The first one is obvious regarding to the second one \(\frac{1}{4}\)=0.25 < \(\frac{1}{3}\)=0.3333.

**Given:**

Lindy’s surfboard wide = \(23 \frac{1}{3}\)

**To find:**

Between which two surfboards model is her custom surfboard’s width? How do you know?

Lindy’s surfboard wide which is also considered as a width of the surfboard with \(23 \frac{1}{3}\).

Though the thickness is not given we can never bother about it.

\(\frac{70}{3}\) is the width of the surfboard We changed the mixed fraction to a rational number because it is easy to understand.

Model 82 and Model 92 is the two surfboard models of Calvin, compared as wide which is approximately the width of Lindy’s surfboard.

**Question. Find how Juanita write the fastball statistic in decimal form **\(\frac{240}{384}\)=0.625.

**Given:**

To find how Juanita write the fastball statistic in decimal form

\(\frac{240}{384}\)=0.625

There were 384 pitches and out of that number 240 were fastballs

This is terminating decimal because after that number five comes infinite number of zeros, none other numbers will appear again.

The answer 0.625 is the decimal form of the fastball statistic

**Question. Determine whether it is terminates or non terminates.**

**Given:**

To determine whether it is terminates or non terminates.

1.\(\frac{100}{3}\)

\(\frac{100}{3}\)=\(33 . \overline{3}\)

It is repeating decimal numbers, there is infinite numbers after the decimal point

2. \(\frac{100}{5}\)

\(\frac{100}{5}\)= 20

This is actually an integer, there are only zeros after the decimal point

3.\(\frac{100}{6}\)=

\(\frac{100}{3}\)=\(33 . \overline{3}\)

It is repeating decimal numbers, there is infinite numbers after the decimal point

\(\frac{100}{3}\)=\(33 . \overline{3}\), \(\frac{100}{6}\)\(16 . \overline{6}\) Are repeating decimals. \(\frac{100}{5}\)= 20 Terminating decimals.

**Question. Explain the given numbers **–\(0.\overline{3}\), **3.1414414441444 are rational numbers or not.**

**Given:**

To explain the given numbers −\(0.\overline{3}\), 3.1414414441444 are rational numbers or not.

First off, we need to know what a rational number is

A rational number is any number that can be made by dividing 2 integers.

**For example:**

1.5 would be a rational number because 1.5=3/2 (3 and 2 are both integers) With that being said, let’s begin.

First off, let’s start with −0.3, let’s ask ourselves, can we make this number by dividing 2 integers?

The answer is yes, we can, by doing we have the equivalent of −0.3, So now we know it’s a rational number.

As for the second one, it can’t be a rational number because it can’t be made by dividing to integers, so the answer is:

−0.3 Is a rational number, but 3.14144144414444 is not.

-0.3 is a rational number, but 3.14144144414444 is not.

**Reasoning:**

In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers. A numerator p and a non-zero denominator q.

**Given:**

Total pitches = 384

Fastball used times = 240

**To find:**

What decimal should Juanita use to update her report?

Times of fastball used divided by the total pitches

\(\begin{array}{r}3 8 4 \longdiv { 2 4 0 0 0 } \\

\frac{-2304}{960} \\

\frac{-768}{1920} \\

\frac{-1920}{0}

\end{array}\)

Therefore, Juanita should use the decimal 0.6205 to update her report.

**Question. Write how rational numbers written as decimals.**

**Given:**

To write how rational numbers written as decimals

Rational numbers written as decimals

Any terminating or repeating decimal that can be written as a fraction using algebraic methods

An integer can be written as a fraction simply by giving it a denominator so any integer is a rational number.

Rational numbers become decimals after we divide the numerator with the denominator

Rational numbers become decimals after we divide numerators with denominators.

**Question. Write how can you use division to find the decimal equivalent of a rational number.**

**Given:**

To write how can you use division to find the decimal equivalent of a rational number.

How can you use division to find the decimal equivalent of a rational number

The decimal forms of rational numbers either end or repeat a pattern.

To convert fractions to decimals you just divide the top by the bottom divide the numerator by the denominator and if the division doesn’t come out evenly.

You can stop after a certain number of decimal places and round off.

The answer is the numerator divided by the denominator.

**Question. Write the difference between a terminating decimal and a repeating decimal number.**

**Given:**

To write the difference between a terminating decimal and a repeating decimal number.

Difference between a terminating decimal and a repeating decimal number.

Terminating decimal ends that means we know how many digits are after the decimal points.

On the other hand, repeating decimals has an infinite number of repeating digits after the decimal points.

The difference only is number of digits.

**Question. Write the decimal equivalent of the rational number** \(\frac{7}{20}\).

**Given:**

To write the decimal equivalent of the rational number \(\frac{7}{20}\).

\(\frac{7}{20}\), dividing both the number in two table.

\(\frac{7}{20}\)=\(\frac{3.5}{10}\)

= 0.35

The answer is 0.35.

**Given:
**

To write the decimal equivalent of the rational number

\(\frac{-23}{20}\).

\(\frac{-23}{20}\), dividing both the number in two table.

\(\frac{-23}{20}\)=\(\frac{-1.15}{20}\)

= -1.15

The answer is -1.15.

**Given:
**

To write the decimal equivalent of the rational number \(\frac{1}{18}\)

\(\frac{1}{18}\) dividing both the number in two table.

\(\frac{1}{18}\) =\(\frac{0.5}{9}\)

= 0.0555

\(0.0 \overline{5}\) (Repeating)

The answer=\(0.0 \overline{5}\) (Repeating).

**Given:**

To write the decimal equivalent of the rational number\(\backslash\left[\frac{-60}{22} \backslash\right]\)

\(\frac{-60}{22}\), Dividing both the number in two tables

\(\frac{-60}{22}\) =\(\frac{-30}{11}\)

= -2.7272

\(2. \overline{72}\) (Repeating)

The answer =\(2. \overline{72}\) (Repeating)

**Question. A mile has 5280 feet there are 1000 feet left to pass, what part of a mile in decimal forms is left to pass? Find what part of a mile in decimal form, will you drive until you reach the exit?**

**Given: **

To find what part of a mile in decimal form, will you drive until you reach the exit?

A mile has 5280 feet there are1000 feet left to pass, what part of a mile in decimal form is left to pass?

\(\frac{1000}{5280}\)=\(0. \overline{1893}\)

The answer =\(0. \overline{1893}\) part of a mile in decimal form is left to pass.

**Question. Write the decimal equivalent for each rational number **\(\frac{2}{3}\).

**Given:**

To write the decimal equivalent for each rational number \(\frac{2}{3}\)

\(\frac{2}{3}\)Canceling the number using 3 table

\(\frac{2}{3}\)=\(0. \overline{6}\)

\(0. \overline{6}\) (Repeating)

The answer \( is 0. \overline{6}\) (Repeating )

**Question. Write the decimal equivalent for each rational number **\(\frac{3}{11}\).

**Given:**

To write the decimal equivalent for each rational number \(\frac{3}{11}\)

\(\frac{3}{11}\)Canceling the number using 11 table

\(\frac{3}{11}\)=\(0. \overline{27}\)

\(0. \overline{27}\) (Repeating)

The answer \( is 0. \overline{27}\) (Repeating )

**Question. Write the decimal equivalent for each rational number **\(8 \frac{4}{9}\).

**Given:**

To write the decimal equivalent for each rational number \(8 \frac{4}{9}\)

\(8 \frac{4}{9}\)Converting the mixed fraction into improper fraction

\(8 \frac{4}{9}\)=\(\frac{76}{9}\)

\(\frac{76}{9}\)=\(8. \overline{4}\) (Repeating)

The answer \(8 \frac{4}{9}\)= \(8. \overline{4}\) (Repeating)

**Question. Write whether **\(1.02 \overline{27}\) **a rational number and also explain it.**

**Given:
**

To write whether \(1.02 \overline{27}\) a rational number and also explain it.

Yes, it is a rational number.

**Explanation:**

The given number is a rational number because it can also be written as fraction and it has also repeating digits.

The given number \(1.02 \overline{27}\) is a rational number.

**Question. Whether the fraction is terminating or not terminating **\(\frac{1}{3}\).

**Given:
**

To tell whether the fraction is terminating or not terminating.

**Answer:**

The given fraction is nonterminate

\(\frac{1}{3}\)=\(0. \overline{3}\)

Which means the \(\frac{1}{3}\) fraction has the repeating decimal.

The answer is given fraction \(\frac{1}{3}\) is repeating decimal.

**Question. Whether the number -34 is a rational number or whole number or integer.**

**Given:**

To tell whether the number −34 is a rational number or whole number or integer.

-34

**Answer:**

1. The given number −34 is not a whole number because it has a negative sign.

2. The given number can be considered as rational number because it has been written in a fractional form \(\frac{-34}{1}\).

3. The given number−34 is also a integer because the given number is a whole number with a negative sign.

The answer −34 is an integer, a rational number but not a whole number.

**Question. Convert decimal to fraction **\(2 \frac{5}{8}\).

**Given:
**

To convert \(2 \frac{5}{8}\) into decimal.

To convert decimal to fraction

\(2 \frac{5}{8}\)=\(\frac{21}{8}\)

2.625

**Given:**

To tell what was ariel’s likely error.

Ariel takes the value of \(\frac{5}{8}\) =0.58 it is the error of ariel

**Explanation :**

We are given that Ariel incorrectly the value of

\(2 \frac{5}{8}\)=2.58

We have to convert \(2 \frac{5}{8}\) into a decimal and we have to find the arials error

\(2 \frac{5}{8}\)=2+\(\frac{5}{8}\)

= 2 + 0.625

= 2.625

\(2 \frac{5}{8}\)=2.625 not 2.58

Ariel consider that value \( \frac{5}{8}\)

= 0.58

But actually the value is 0.625

Hence ariel takes value of 0.58 in place of0.625

**Solution:**

Ariel takes the value of \(\frac{5}{8}\) =0.58 it is the error of ariel.

**Question. Find the value of a and b in a√b when you use division to find the decimal form. Find the decimal form we have to divide 3 by 11.**

**Given:**

To find the value of a and b in a√b when you use division to find the decimal form.

To find the decimal form we have to divide 3 by 11

a√b

a=?

b=?

b is numerator = 3

a is denominator = 11

Solution: a = 11,b = 3

**Given:
**

To find decimal form for \(\frac{3}{11}\)

To find the decimal form \(\frac{3}{11}\)

Canceling the terms with 11 table

\(\frac{3}{11}\)=\(0. \overline{27}\).

**Question. Find the decimal should the digital scale. Daniel wants to find **\(3 \frac{1}{5}\) **lb of ham.**

**Given:
**

To find what decimal should the digital scale.

Daniel wants to find \(3 \frac{1}{5}\) lb of ham

First, we have to change the mixed fraction to a normal fraction.

The scale shoulder read 3.2 lb

The scale should read 3.2 lb

**Question. Find number of pounds of port she brought using a decimal **\(18 \frac{8}{25}\) **is a mixed fraction.**

**Given:**

To find number of pounds of port she brought using a decimal.

\(18 \frac{8}{25}\) is a mixed fraction

In order to convert the mixed fraction to decimal form you need to divide the numerator by the denominator

\(\frac{8}{25}\) = 0.32

Next add the decimal\(\frac{8}{25}\) =0.32 to the whole number from the mixed fraction

18+0.32 = 18.32

This is how many pounds she bought .this decimal terminates.

The 18.32 number of pounds of pork she brought.

**Question. Explain the given number 9.373 is a repeating decimal or not.**

**Given:**

To explain the given number 9.373 is a repeating decimal.

9.373

It is not a repeating decimal

**Reason:**

There are no repeating digits,9.373 the number has only three digit after the decimal point.

It is a rational because rationals are all numbers that can be written as fractions.

The given number 9.373 is not a repeating decimal.

**Question. Find how tall will the stack.**

**Given:**

To find how tall will the stack be.

First box=\(3 \frac{3}{11}\)

Second box = 3.27

We have to convert the\(3 \frac{3}{11}\) into the normal form

\(3 \frac{3}{11}\) = 0.27

\(3 \frac{3}{11}\) = 3 + 0.27

3.27

If he stacks both the boxes

3.27 + 3.27 = 6.54

The stack height will be 6.54.

**Question. Find the decimal should you watch for on the digital pressure gauge.**

**Given:
**

To find what the decimal should you watch for on the digital pressure gauge.

We should watch the dia0.1 on the pressure gauge

The air pressure in the tyre \(32 \frac{27}{200}\)

We need to convert the mixed fraction to decimal number

\(\frac{27}{200}\) = 0.135

\(32 \frac{27}{200}\) = 32 + 0.135

= 32.135

Thus the air pressure in the tyre = 32.135 pounds per square inch

Rounding to the nearest tenths we get 32.1 pounds per square inch

**Solution: **

The air pressure in the tyre = 32.135 pounds per square inch.

**Question. Justify that the pizza should fit to the square box.**

**Given:
**

To justify that the pizza should fit to the square box.

**Answer:** The pizza fit in the box

**Explanation:**

The diameter of the pizza is D = \(10 \frac{1}{3}\)

The wide of the square box is b = 10.38

Wkt

For the pizza to fit inside the box, it must be fulfilled that

D ≤ b

We need to convert the decimal to a fraction

\(10.38\left(\frac{100}{100}\right)=\frac{1038}{100}\)= \(\frac{519}{50}\)

Convert \(10 \frac{1}{3}\) into improper fraction

\(10 \frac{1}{3}=\frac{31}{3}\)Multiply the diameter of the pizza by 50 both numerator and denominator

\(\frac{31}{3} \times \frac{50}{50}=\frac{1550}{150}\)Multiply the wide of the box 3 in both numerator and denominator

\(\frac{519}{50} \times \frac{3}{3}\)=\(\frac{1557}{150}\)

\(\frac{1557}{150}<\frac{1550}{150}\)Therefore the pizza fit inside the box.

The pizza will fit inside the box.

**Question. Convert the fraction to decimal choose the correct answer for **\(117 \frac{151}{200}\).

**Given:
**

To choose the correct answer for \(117 \frac{151}{200}\)

**Answer:**

Convert the fraction to decimal

\(\frac{151}{200}\) =0.755

\(117 \frac{151}{200}\) =117+0.755

117.055

117.055 is correct answer.

**Question. Convert the fraction to decimal find the decimal equivalents for each fraction for **\(\frac{-4}{5}, \frac{-5}{6}\).

**Given:
**

To find the decimal equivalents for each fraction for \(\frac{-4}{5}, \frac{-5}{6}\)

To convert the fraction to decimal

\(\frac{-4}{5}\) = -0.8

\(\frac{-5}{6}\) = –\(0.8 \overline{3}\)

The decimal equivalent for \(\frac{-4}{5}\),\(\frac{-5}{6}\)

= −0.8,-\(0.8 \overline{3}\)

**Given:**

Which is a repeating decimal? which digit is repeating?

To find the repeating decimal.

\(\frac{-4}{5}\) = -0.8

\(\frac{-5}{6}\) = -0.8333……

-0.8333…… is the repeating decimal and the number 3 is repeating.