Additional Arithmetic Rules: Number of terms

Recall the order of operations: Parentheses precede all other operations, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right)

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, hence 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 it we'll remove the parentheses entirely:

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

Once again we'll perform the operation in the remaining parentheses and after calculating it we'll remove the parentheses entirely:

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

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

Shown below is the solution that we obtained expressed as a fraction. Proceed 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, 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:

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

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:

$3+0.15=3.15$

Adhering to the order of operations: First calculate what's inside of the parentheses, proceed to the multiplication and division operations (left to right) and finally perform the addition and subtraction operations. (left to right)

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, after which 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 in fact 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 after which 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 begin by solving the expression inside of the parentheses:

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

Let's write the division problem as a fraction:

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

Given that this is a fraction divided by a fraction, we'll proceed to combine everything into one fraction:

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

Break down 24 into a multiplication of:

$8\times3$

As seen below in the fraction's numerator:

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

Proceed to reduce the 3 in both the numerator and denominator as follows:

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

Swap the numerator with the denominator, resulting in a multiplication problem where the denominator is 8:

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

Combine the 12 with the fraction's numerator given that this is a simple multiplication operation:

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

Proceed to break down the 12 in the numerator and the 8 in the denominator into multiplication problems:

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

Reduce the 4 in both the numerator and denominator:

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

Solve the expression in the numerator from left to right:

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

We obtain the following fraction:

$\frac{135}{2}$

Break down the fraction into an addition of fractions:

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

Proceed to solve both fractions:

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

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

We obtain the following expression:

$65+2.5=67.5$

Recall the order of operations: parentheses precede all other operations, proceed to perform multiplication and division (from left to right) finally address the addition and subtraction operations (from left to right)

When there are parentheses within parentheses, 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, after which we can remove the inner parentheses. We'll continue doing this until no more parentheses remain in the exercise.

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

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

Express the value obtained from the inner parentheses as a fraction and proceed to reduce it.

Reminder - How do we approach reducing a fraction? Divide both the 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?

In order to determine 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 -

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

Recall the order of operations: parentheses precede all other operations, next precede to the multiplication and division operations (from left to right), addition and subtraction come last (from left to right)

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, after which we can remove the inner parentheses. We'll continue doing this until no more parentheses remain in the exercise.

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

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

Express the value obtained in the inner parentheses as a fraction and precede to reduce it.

Reminder - how do we approach fraction reduction? Divide both the 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

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

Remember that division by definition is in fact 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)

Recall the order of operations: parentheses precede all other operations, multiplication and division follow (from left to right) and finally addition and subtraction (from left to right)

When there are parentheses within parentheses, we start with the innermost ones first.

$3:(6\times(12:(2\operatorname{\times}3)))=$

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

Therefore, we will first perform the operation in the 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.

$3:(6\times(12:(2\times3)))=3:(6\times(12:(6))) =$

$3:(6\times(12:(6))) =3:(6\times(12:6)) =$

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

$3:(6\times(12:6)) =3:(6\times(2)) =$
$3:(6\times(2))=3:(6\times2)=$

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

$3:(6\times2)= 3:(12)=$

$3:(12)= 3:12 =$

Shown below is the solution that we obtained expressed as a fraction. Proceed to reduce it.

$3:12=\frac{3}{12}=$

The largest number by which we can reduce the fraction is 3:

$\frac{3}{12}=\frac{3:3}{12:3}=$

Proceed to solve it:

$\frac{3:3}{12:3}=\frac{1}{4}$

Therefore the answer is option c -

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

Adhering to the order of operations: Begin by calculating what's inside of the parentheses, then proceed to multiplication and division (from left to right), finish with addition and subtraction (from left to right)

When there are parentheses within parentheses, we start with the innermost ones first.

$-38-(24-(17-5)) =$

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

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

$-38-(24-(17-5)) = -38-(24-(12))$

$-38-(24-(12)) = -38-(24-12)$

Now we'll perform the operation in the remaining parentheses. We'll then proceed to remove the parentheses

$-38-(24-12) = -38-(12)$

$-38-(12) = -38-12$

Reminder - Addition and subtraction of numbers with the same sign

When we have two numbers with the same sign (plus or minus), that sign will remain in the result, which will actually be the result of an addition exercise.

$-38-12 = -50$

Therefore the answer is option c - (-50)

Whilst adhering to the order of operations: parentheses precede multiplication and division (from left to right) which in turn precede addition and subtraction (from left to right)

When parentheses within parentheses are present, we start with the innermost ones first.

$72:(7\times(24:(4\operatorname{\times}2)))=$

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 are able to remove the inner parentheses. We'll continue doing this until no more parentheses remain in the exercise.

$72:(7\times(24:(4\operatorname{\times}2)))=72:(7\times(24:(8)))=$

$72:(7\times(24:(8)))=72:(7\times(24:8))=$

$72:(7\times(24:8))=72:(7\times(3))=$

$72:(7\times(3))=72:(7\times3)=$

$72:(7\times3)=72:(21)=$

Express the resulting numerical value obtained from the inner parentheses as a fraction and then proceed to reduce the fraction.

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

$72:(21)=72:21=\frac{72}{21}=$
The largest number by which we can reduce the fraction is 3

$\frac{72}{21}=\frac{72:3}{21:3}=$

$\frac{72:3}{21:3}=\frac{24}{7}=$

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

In order to determine the numerator - check how many times the denominator goes into the numerator and add the remainder to the numerator.

For the denominator – don't change anything.

In this exercise, the numerator is 24 and the denominator is 7

7 goes into 24 three times with a remainder of 3

$\frac{24}{7}=3\text{ }\frac{3}{7}$

Therefore the answer is option a -

$3\text{ }\frac{3}{7}$

Adhering to the order of operations: First calculate any operations inside of the parentheses. Next proceed to the multiplication and division operations (left to right) and finally address the addition and subtraction operations.(left to right)

When there are parentheses within parentheses, we start with the innermost ones first.

$73-(89-(37-45))=$

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

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

$73-(89-(37-45))=73-(89-(-8))$

Pay attention

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

When there is a sequence of two plus signs, the result will be plus.

When there is a sequence of two minus signs, the result will be plus.

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

$73-(89-(-8)) = 73-(89+8)$
Now we'll perform the operation in the remaining parentheses and after calculating we'll eventually remove the parentheses

$73-(89+8)= 73-(97)$

$73-(97) = 73-97$

Reminder - Adding and Subtracting Signed 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 problem.

$73-97 = -24$

Therefore the answer is option c - (-24)

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 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)