Question

if two balls have the same volume, but ball a has twice as much mass as ball b, which one will have the greater density?
if ball c is 3 times the volume of ball d and ball d has 1/3 the mass of ball c, which has the greater density?
if two balls have the same mass, but ball p is twice as large as ball q, which one will have the greater density?
if ball x is twice as big as ball y and weighs only half as much as ball y, then which one will have the greater density?

Explanation:

Step1: Recall the density formula

Density formula is \(
ho=\frac{m}{V}\), where \(
ho\) is density, \(m\) is mass and \(V\) is volume.

Step2: Analyze the first case

Let \(V_A = V_B=V\), \(m_A = 2m_B\). Then \(
ho_A=\frac{m_A}{V_A}=\frac{2m_B}{V}\), \(
ho_B=\frac{m_B}{V}\). Since \(\frac{2m_B}{V}>\frac{m_B}{V}\), ball A has greater density.

Step3: Analyze the second case

Let \(V_C = 3V_D\), \(m_D=\frac{1}{3}m_C\). Then \(
ho_C=\frac{m_C}{V_C}=\frac{m_C}{3V_D}\), \(
ho_D=\frac{m_D}{V_D}=\frac{\frac{1}{3}m_C}{V_D}=\frac{m_C}{3V_D}\). So \(
ho_C=
ho_D\)

Step4: Analyze the third case

Let \(m_P = m_Q = m\), \(V_P=2V_Q\). Then \(
ho_P=\frac{m}{V_P}=\frac{m}{2V_Q}\), \(
ho_Q=\frac{m}{V_Q}\). Since \(\frac{m}{V_Q}>\frac{m}{2V_Q}\), ball Q has greater density.

Step5: Analyze the fourth case

Let \(V_X = 2V_Y\), \(m_X=\frac{1}{2}m_Y\). Then \(
ho_X=\frac{m_X}{V_X}=\frac{\frac{1}{2}m_Y}{2V_Y}=\frac{m_Y}{4V_Y}\), \(
ho_Y=\frac{m_Y}{V_Y}\). Since \(\frac{m_Y}{V_Y}>\frac{m_Y}{4V_Y}\), ball Y has greater density.

Answer:

Ball A has greater density.
Ball C and ball D have the same density.
Ball Q has greater density.
Ball Y has greater density.