Question

Unknown Quantity Word Problems

Answer 1

We have two equations.

Let M be the Michaela's age and J the Jake's age.

The first one is that Michaela's age is ten years more than half the age of Jake.

`M=(J)/(2)+10`

The second equation can be written knowing that the sum of the two ages is 55:

`M+J=55`

We can start by substituting J in the first equation, using the information of the second equation:

`\begin{gathered} J=55-M \\ M=(J)/(2)+10=((55-M))/(2)+10=(55)/(2)-(M)/(2)+10=(75)/(2)-(M)/(2) \\ M=(75)/(2)-(M)/(2) \\ M+(M)/(2)=(75)/(2) \\ (3)/(2)M=(75)/(2) \\ 3M=75 \\ M=(75)/(3) \\ M=25 \end{gathered}`

The age of Michaela is 25 years.

7) Lets call

H: the number of atoms of hydrogen

C: the number of atoms of carbon

O: number of atoms of oxygen

We know that there are twice as many atoms of hydrogen as oxygen. This means that if we have 2 oxygens, we should have 4 hydrogens.

This can be written as:

`H=2O`

We know that one more atom of carbon than hydrogen.

This is:

`C=H+1`

We know that the sum of atoms is 21. This is:

`H+C+O=21`

Now that we have 3 equations for the three unknowns we start by solving for C.

We use the substitution method to solve it.

`C=H+1\longrightarrow H=C-1`
`H=2O=C-1\longrightarrow O=(C)/(2)-(1)/(2)`

Then, we can replace H and O in the 3rd equation:

`\begin{gathered} H+C+O=21 \\ (C-1)+C+((C)/(2)-(1)/(2))=21 \\ (1+1+(1)/(2))C-1-(1)/(2)=21 \\ (5)/(2)C-(3)/(2)=21 \\ (5)/(2)C=21+(3)/(2)=(42+3)/(2)=(45)/(2) \\ (5C)/(2)=(45)/(2) \\ C=(45)/(2)\cdot(2)/(5)=9 \end{gathered}`

The numbers of atoms of carbon is 9 atoms.

Alternative method:

Atoms of hydrogen = 2 * atoms of oxygen

Atoms of hydorgen = atoms of carbon - 1

Then we can say that:

Atoms of hydrogen = 2 * atoms of oxygen = atoms of carbon -1

Then, from the last equation:

Atoms of oxygen = (1/2)*atoms of carbon - (1/2)

If we add all the atoms, we get 21 atoms:

Atoms of hydrogen + Atoms of oxygen +Atoms of carbon = 21

We will replace each of the types of atoms for an expression where the atoms of carbon are present, because is what we need for our answer.

We will call x the number of atoms of carbon.

As:

Atoms of hydrogen = atoms of carbon - 1 = x-1

Atoms of oxygen = (1/2)*atoms of carbon - (1/2) = 1/2*x-1/2

We will have:

`\begin{gathered} Atomsofhydrogen+Atomsofoxygen+Atomsofcarbon=21 \\ (x-1)+((1)/(2)x-(1)/(2))+x=21 \\ x-1+(1)/(2)x-(1)/(2)+x=21 \\ (x+(1)/(2)x+x)-1-(1)/(2)=21 \\ (5)/(2)x-(3)/(2)=21 \\ (5)/(2)x=21+(3)/(2)=(21\cdot2+3)/(2)=(42+3)/(2)=(45)/(2) \\ x=(45)/(2)\cdot(2)/(5)=9 \end{gathered}`

x, the number of atoms of carbon, is equal to 9.