Additional Arithmetic Rules: Number of terms

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$15/(4:\frac{2}{8})=$

Now we invert the fraction to create a multiplication exercise:

$15/(4\times\frac{8}{2})=$

We add the 4 to the numerator of the fraction in the multiplication exercise:

$15/(\frac{4\times8}{2})=$

Now we invert the fraction to create a multiplication exercise:

$15\times\frac{2}{4\times8}=$

We add the 15 to the numerator of the fraction in the multiplication exercise:

$\frac{15\times2}{4\times8}=$

We separate the 4 into a smaller multiplication exercise:

$\frac{15\times2}{2\times2\times8}=$

We simplify the 2 in the numerator and denominator:

$\frac{15}{2\times8}=\frac{15}{16}$

Let's recall the order of operations: calculate what's in parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We emphasize that when there are parentheses within parentheses, we start with the innermost ones first.

$24:(2\times(16\times(2+1)))=$

In this exercise, there are addition operations in the inner parentheses, and the rest of the exercise includes multiplication and division operations and parentheses within parentheses.

Therefore, we will first perform the operation in the inner parentheses, and after calculating we can remove the inner parentheses. We'll continue doing this until there are no more parentheses in the exercise.

At each step in solving operations within parentheses, we'll make sure to follow the order of operations

$24:(2\times(16\times(2+1)))= 24:(2\times(16\times(3)))=$

$24:(2\times(16\times(3)))= 24:(2\times(16\times3))=$

Note - In the inner parentheses there was only addition operation, so we followed the order of operations when we solved this operation first and not the multiplication and division operations

Now we'll perform the operation in the remaining parentheses and after calculating we'll remove the parentheses

$24:(2\times(16\times3))=24:(2\times(48)) =$
$24:(2\times(48)) =24:(2\times48) =$

And again we'll perform the operation in the remaining parentheses and after calculating we'll remove the parentheses

$24:(2\times48) = 24:(96) =$

$24:(96) = 24:96 =$

Let's show the solution we got as a fraction and try to reduce it

$24:96=\frac{24}{96}=$

The largest number by which we can reduce the fraction is 24

$\frac{24}{96}=\frac{24:24}{96:24}=$

Let's solve

$\frac{24:24}{96:24}= \frac{1}{4}$

Therefore the answer is section a -

$(\frac{1}{4})$

First, let's write the multiplication exercise in the inner parentheses as a fraction:

$30/(3:\frac{13}{2})=$

Now let's flip the fraction to create a multiplication exercise:

$30/(3\times\frac{2}{13})=$

Let's add 13 to the fraction's numerator in a multiplication exercise:

$30/(\frac{3\times2}{13})=$

Now let's flip the fraction to create a multiplication exercise:

$30\times\frac{13}{3\times2}=$

Let's add 30 to the fraction's numerator in a multiplication exercise:

$\frac{30\times13}{3\times2}=$

Let's break down the 30 into a smaller multiplication exercise:

$\frac{10\times3\times13}{3\times2}=$

Let's reduce between the 3 in the numerator and denominator:

$\frac{10\times13}{2}=$

Let's break down the 10 into a smaller multiplication exercise:

$\frac{5\times2\times13}{2}=$

Let's reduce between the 2 in the numerator and denominator to get:

$5\times13=65$

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$3:(4:\frac{5}{12})=$

Now we invert the fraction to create a multiplication exercise:

$3:(4\times\frac{12}{5})=$

We add the 4 to the numerator of the fraction in the multiplication exercise:

$3:\frac{4\times12}{5}=$

Now we invert the fraction to create a multiplication exercise:

$3\times\frac{5}{4\times12}=$

We add the 3 to the numerator of the fraction in the multiplication exercise:

$\frac{3\times5}{4\times12}=$

We break down the 12 into a smaller multiplication exercise:

$\frac{3\times5}{4\times3\times4}=$

We simplify the 3 in the numerator and denominator:

$\frac{5}{4\times4}=\frac{5}{16}$

Let's recall the order of arithmetic operations: calculate what's in parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We emphasize that when there are parentheses within parentheses, we start with the innermost ones first.

$43-(35-(24-18))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation in the inner parentheses, and after calculating we can remove the inner parentheses and be left with only one set of parentheses.

$43-(35-(24-18))= 43-(35-(6))$

$43-(35-(6))=43-(35-6)$
Now we will perform the operation in the remaining parentheses and after calculating we will remove the parentheses

$43-(35-6) = 43-(29)$

$43-(29) = 43-29 = 14$

Therefore the answer is option a - (14)

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$7-(10-(4-3))=$
In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$7-(10-(1))=7-(10-1)=$

When we perform an operation of ("addition and subtraction") with directed numbers, we enclose the directed number inside of parentheses.

Parentheses can be omitted, but when omitting the parentheses, remember that (-=+-)

$7-(9)=7-9=-2$

Reminder: addition and subtraction of directed numbers

In this case, we remember that when we have two numbers with different signs, it is important to determine which number is greater in terms of absolute value (absolute - the distance from zero). The larger number will determine the sign of the result and we will perform a subtraction exercise.

Therefore, the correct answer is option c: (-2)

First, we rewrite the multiplication exercise within the inner parentheses as a fraction:

$75:(8:\frac{4}{15})=$

Now we invert the fraction to create a multiplication exercise:

$75:(8\times\frac{15}{4})=$

We add the 8 to the numerator of the fraction in the multiplication exercise:

$75:\frac{8\times15}{4}=$

Now we invert the fraction to create a multiplication exercise:

$75\times\frac{4}{8\times15}=$

We add the 75 to the numerator of the fraction in the multiplication exercise:

$\frac{75\times4}{8\times15}=$

We break down the 75 and the 8 into smaller multiplication exercises:

$\frac{15\times5\times4}{4\times2\times15}=$

We simplify the 4 and the 15 in the numerator and denominator:

$\frac{5}{2}=2\frac{1}{2}$

Let's look at the expression in parentheses and write it as a fraction:

$(8\times(10:(3\times12)))=8\times\frac{10}{3\times12}$

Now we'll get the expression:

$7:(8\times\frac{10}{3\times12})=$

Let's address the parentheses and combine the 8 with the multiplication in the numerator:

$7:(\frac{8\times10}{3\times12})=$

Let's break down the 8 and 12 into smaller multiplication problems:

$7:(\frac{4\times2\times10}{3\times4\times3})=$

Let's reduce between the 4 in the numerator and denominator and get:

$7:(\frac{2\times10}{3\times3})=$

Let's solve the multiplication problems in the parentheses and get:

$7:(\frac{20}{9})=$

Let's switch between the numerator and denominator so we can turn the expression into multiplication and add the 7 to the fraction's numerator:

$7\times\frac{9}{20}=\frac{7\times9}{20}=\frac{63}{20}$

Let's separate the fraction's numerator into an addition problem:

$\frac{60+3}{20}=$

Now let's separate it into an addition of fractions:

$\frac{60}{20}+\frac{3}{20}=3+\frac{3}{20}=$

Let's multiply the fraction by 5:

$3+\frac{3\times5}{20\times5}=3+\frac{15}{100}$

And we'll get the expression:

$3+0.15=3.15$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$10-(12-(4-12))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$10-(12-(4-12))=10-(12-(-8))$

Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$10-(12-(-8)) = 10-(12+8)$

Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$10-(12+8)=10-(20)$$10-\left(20\right)=10-20=-10$

Therefore, the correct answer is option a: (-10)

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$10:(2:\frac{15}{7})=$

Now we invert the fraction to create a multiplication exercise:

$10:(2\times\frac{7}{15})=$

We add the 2 to the numerator of the fraction in the multiplication exercise:

$10:\frac{2\times7}{15}=$

Now we invert the fraction to create a multiplication exercise:

$10\times\frac{15}{2\times7}=$

We add the 10 to the numerator of the fraction in the multiplication exercise:

$\frac{10\times15}{2\times7}=$

We break down the 10 into a smaller multiplication exercise.

$\frac{5\times2\times15}{2\times7}=$

We simplify the 2 in the numerator and denominator:

$\frac{5\times15}{7}=\frac{75}{7}$

We separate the fraction into a sum exercise between fractions:

$\frac{70+5}{7}=\frac{70}{7}+\frac{5}{7}=10+\frac{5}{7}=10\frac{5}{7}$

We rewrite the innermost parentheses in fraction form:

$10/(7:\frac{9}{2})=$

We convert the parentheses into a multiplication exercise through inverting the fraction:

$10/(7\times\frac{2}{9})=$

We add the 7 to the numerator for the multiplication exercise:

$10/(\frac{7\times2}{9})=10:\frac{7\times2}{9}$

We convert the exercise into a multiplication by inverting the fraction:

$10\times\frac{9}{7\times2}=$

We add the 10 to the numerator for the multiplication exercise:

$\frac{10\times9}{7\times2}=$

We break down the 10 into a simpler multiplication exercise:

$\frac{5\times2\times9}{7\times2}=$

We simplify the 2 in the numerator and denominator:

$\frac{5\times9}{7}=\frac{45}{7}$

We convert the fraction's numerator into a sum exercise:

$\frac{42+3}{7}=\frac{42}{7}+\frac{3}{7}=6+\frac{3}{7}=6\frac{3}{7}$

Let's recall the order of operations: calculate what's in parentheses, multiplication and division (left to right), addition and subtraction (left to right)

We emphasize that when there are parentheses within parentheses, we start with the innermost ones first.

$108-(-46-(3-24)) =$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation in the innermost parentheses, and after calculating we can remove the inner parentheses. We'll continue doing this until there are no more parentheses in the exercise.

Reminder - Addition and Subtraction of Directed Numbers

When we have two numbers with different signs, it's important to determine which number is larger in absolute value (absolute - distance from zero). The larger number will determine the sign of the result, and we'll actually perform a subtraction exercise.

$108-(-46-(3-24)) = 108-(-46-(-21)) =$

Pay Attention

When an exercise contains a sequence of two signs (which are usually separated by parentheses) we'll distinguish between several cases:

When there's a sequence of two plus signs, the result will also be plus.

When there's a sequence of two minus signs, the result will also be plus.

When there's a sequence of minus and plus or plus and minus, the result will be minus. 

$108-(-46-(-21)) =108-(-46+21) =$

Now we'll perform the operation in the remaining parentheses and after calculating we'll remove the parentheses

$108-(-46+21) = 108-(-25) =$
$108-(-25) = 108+25 = 133$

Therefore the answer is option d - (133)

According to the order of operations rules, we'll first solve the expression in parentheses:

$(\frac{1}{3}\times(24:(15\times3))=$

Let's write the division problem as a fraction:

$\frac{1}{3}\times\frac{24}{15\times3}=$

Since this is a fraction divided by a fraction, we'll combine everything into one fraction:

$\frac{1\times24}{3\times15\times3}=$

Let's break down 24 into a multiplication of:

$8\times3$

And we'll get in the fraction's numerator:

$\frac{1\times8\times3}{3\times15\times3}=$

We'll reduce the 3 in both numerator and denominator and get the expression:

$12:\frac{8}{15\times3}=$

We'll swap between numerator and denominator, so we get a multiplication problem where the denominator is 8:

$12\times\frac{15\times3}{8}=$

We'll combine the 12 with the fraction's numerator since this is just a multiplication operation:

$\frac{12\times15\times3}{8}=$

Let's break down the 12 in the numerator and the 8 in the denominator into multiplication problems:

$\frac{4\times3\times15\times3}{4\times2}=$

We'll reduce the 4 in both numerator and denominator:

$\frac{3\times15\times3}{2}=$

Let's solve the expression in the numerator from left to right:

$3\times15\times3=45\times3=135$

We'll get the fraction:

$\frac{135}{2}$

Let's break down the fraction into an addition of fractions:

$\frac{130}{2}+\frac{5}{2}=$

Let's solve both fractions:

$\frac{130}{2}=130:2=65$

$\frac{5}{2}=5:2=2.5$

Now we'll get the expression:

$65+2.5=67.5$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$13-(10-((-4)-(-10))) =$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$13-(10-((-4)-(-10))) = 13-(10-((-4)+10))$

Reminder: addition and subtraction of directed numbers

When we have two numbers with different signs, it is important to determine which number is greater in terms of absolute value (absolute - the distance from zero). The larger number will determine the sign of the result and we will perform a subtraction exercise.

Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$13-(4) = 13-4 = 9$

Therefore, the correct answer is c: (9)

Let's first address the expression in parentheses and convert it to a fraction:

$(9\times(4:(18\times5)))=(9\times\frac{4}{18\times5})$

Now we'll get the expression:

$13:(9\times\frac{4}{18\times5})=$

We'll add the 9 to the numerator of the fraction in the multiplication expression, and we'll separate the 18 into a smaller multiplication expression:

$13:(\frac{9\times4}{2\times9\times5})=$

Let's reduce the 9 in both numerator and denominator:

$13:(\frac{4}{2\times5})=$

Let's factor the numerator into a multiplication expression:

$13:(\frac{2\times2}{2\times5})=$

Let's reduce the 2 in both numerator and denominator:

$13:(\frac{2}{5})=$

We'll write the fraction in inverse form so we can convert the expression to a multiplication expression:

$13\times\frac{5}{2}=\frac{13\times5}{2}=\frac{65}{2}$

Let's factor the numerator into an addition expression:

$\frac{65}{2}=\frac{60+5}{2}$

Let's separate it into an addition of fractions:

$\frac{60}{2}+\frac{5}{2}=30+2.5=32.5$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$17-(3-(-7-4))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

Adding and subtracting directed numbers is based on several key principles. In this case, we will remember that when we have two directed numbers with the same sign (plus or minus), the result will also have the same sign, which will actually be the result of an addition exercise.

$17-(3-(-7-4))= 17-(3-(-11))$
Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$17-(3-(-11))=17-(3+11)=$
Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$17-(14)=17-14=3$

Therefore, the correct answer is option b: (3)

First, we rewrite down the multiplication exercise within the inner parentheses as a fraction:$25:(3:\frac{10}{35})=$

Now we invert the fraction to create a multiplication exercise:

$25:(3\times\frac{35}{10})=$

We add the 3 to the numerator of the fraction in the multiplication exercise:

$25:\frac{3\times35}{10}=$

Now we invert the fraction to create a multiplication exercise:

$25\times\frac{10}{3\times35}=$

We add the 25 to the numerator of the fraction in the multiplication exercise:

$\frac{25\times10}{3\times35}=$

We separate the 10 and 35 into smaller multiplication exercises:

$\frac{25\times5\times2}{3\times7\times5}=$

We simplify the 5 in the numerator and denominator:

$\frac{25\times2}{3\times7}=\frac{50}{21}$

We separate the numerator of the fraction into a sum exercise:

$\frac{42+8}{21}=$

We separate into sum exercises among fractions:

$\frac{42+8}{21}=\frac{42}{21}+\frac{8}{21}=2+\frac{8}{21}=2\frac{8}{21}$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$25-(34-(20-8))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$25-(34-(20-8))=25-(34-(12))$
Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$25-(34-(12))=25-(34-12)$
Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$25-(34-12)=25-(22)$

$25-(22)=25-22=3$

Therefore, the correct answer is option c: (3)

Let's recall the order of operations: calculating what's in parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We emphasize that when there are parentheses within parentheses, we start with the innermost ones first.

$2\operatorname{\times}(200:(35\operatorname{\times}3))=$

In this exercise, there are only multiplication and division operations and parentheses within parentheses.

Therefore, we will first perform the operation in the inner parentheses, and after calculating we can remove the inner parentheses. We'll continue doing this until there are no more parentheses in the exercise.

$2\times(200:(35\times3))=2\times(200:(105))=$

$2\times(200:(105))= 2\times(200:105) =$

Let's represent the part of the exercise we got in the inner parentheses as a fraction and try to reduce the fraction

Reminder - How do we approach reducing a fraction? Divide both numerator and denominator by the same number

$2\times(200:105)=2\times(\frac{200}{105})=$
The largest number by which we can reduce the fraction is 5

$2\times(\frac{200}{105})= 2\times(\frac{200:5}{105:5})=$

$2\times(\frac{200:5}{105:5})= 2\times(\frac{40}{21})=$

$(\frac{ 2\times40}{21})= (\frac{ 80}{21})=\frac{ 80}{21}$

Reminder - How do we convert an improper fraction to a mixed number?

To find the numerator - we need to check how many times the denominator goes into the numerator and add the remainder to the numerator.

For the denominator – we don't change anything.

In this exercise, the numerator is 80 and the denominator is 21

21 goes into 80 three times with a remainder of 17

$\frac{ 80}{21}=3\text{ }\frac{17}{21}$

Therefore the answer is option a -

$3\text{ }\frac{17}{21}$

Let's recall the order of operations: calculate what's in parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We emphasize that when there are parentheses within parentheses, we start with the innermost ones first.

$35:(3\times(4:(3\operatorname{\times}8)))=$

In this exercise, there are only multiplication and division operations and parentheses within parentheses.

Therefore, we will first perform the operation in the inner parentheses, and after calculating we can remove the inner parentheses. We'll continue doing this until there are no more parentheses in the exercise.

$35:(3\times(4:(3\operatorname{\times}8)))= 35:(3\times(4:(24)))=$

$35:(3\times(4:(24))) = 35:(3\times(4:24))=$

Let's represent the part of the exercise we got in the inner parentheses as a fraction and try to reduce the fraction

Reminder - how do we approach fraction reduction? Divide both numerator and denominator by the same number

$35:(3\times(4:24))=35:(3\times(\frac{4}{24}))=$
$35:(3\times(\frac{4}{24}))=35:((\frac{3\times4}{24}))=$
$35:((\frac{3\times4}{24}))=35:(\frac{12}{24})=$

The largest number by which we can reduce the fraction is 12

$35:(\frac{12}{24})= 35:(\frac{1}{2})=$

Let's remember that division by definition is actually multiplication by the reciprocal

$\frac{a}{\frac{1}{b}}=a\times b$

This means that in this case, instead of dividing by one-half, we can multiply by two, which is the reciprocal of one-half

$35:(\frac{1}{2})=35\times(\frac{2}{1})=35\times(2)=$

$35\times(2)=35\times2=$

$35\times2=70$

Therefore the answer is option A - (70)