Question

Ringxheart=hathat×2=heartheart-ring=1/4

Answer 1

Let's begin by identifying key information given to us:

Let ring be represented as x

Let heart be represented as y

Let hat be represented as z

`\begin{gathered} ring* heart=hat\Rightarrow x\cdot y=z \\ hat*2=heart\Rightarrow z\cdot2=y\Rightarrow y=2z \\ heart-ring=(1)/(4)\Rightarrow y-x=(1)/(4) \\ x\cdot y=z-----1 \\ y=2z------2 \\ y-x=(1)/(4)----3 \end{gathered}`

We will calculate for the unknown from the equations:

`\begin{gathered} x\cdot y=z------1 \\ y=2z-------2 \\ y-x=(1)/(4)-----3 \\ \text{Substitute equation 2 into }equation\text{ 1:} \\ x(2z)=z \\ \text{Divide both sides by ''z'', we have:} \\ (x(2z))/(z)=(z)/(z)\Rightarrow2x=1 \\ 2x=1 \\ x=(1)/(2) \\ \text{Substitute the value of ''x'' into equation 3, we have:} \\ y-x=(1)/(4)\Rightarrow y-(1)/(2)=(1)/(4) \\ y-(1)/(2)=(1)/(4) \\ y=(1)/(4)+(1)/(2)=(3)/(4) \\ y=(3)/(4) \\ \text{Substitute the value of ''y'' into equation 2, we have:} \\ y=2z \\ (3)/(4)=2z \\ \text{Divide both sides by 2, we have:} \\ (3)/(4\cdot2)=(2z)/(2) \\ (3)/(8)=z \\ z=(3)/(8) \end{gathered}`

Therefore,

ring = 1/2

heart = 3/4

hat = 3/8