Let’s solve the problem step by step.

Given:

1. $50\mathrm{\%}50\backslash \%$ of the greater number = $60\mathrm{\%}60\backslash \%$ of the smaller number.
2. The sum of the two numbers is 150.

Let:

• Greater number = $xx$
• Smaller number = $yy$

Step 1: Express the relationship between $xx$ and $yy$:

From the condition:
$0.5x=0.6y0.5x\; =\; 0.6y$
Dividing both sides by $0.10.1$:
$5x=6y5x\; =\; 6y$
Or:
$x=\frac{6}{5}yx\; =\; \backslash frac\{6\}\{5\}y$

Step 2: Use the sum of the numbers:

From the second condition:
$x+y=150x\; +\; y\; =\; 150$

Substitute $x=\frac{6}{5}yx\; =\; \backslash frac\{6\}\{5\}y$ into $x+y=150x\; +\; y\; =\; 150$:
$\frac{6}{5}y+y=150\backslash frac\{6\}\{5\}y\; +\; y\; =\; 150$
Combine terms:
$\frac{6}{5}y+\frac{5}{5}y=150\backslash frac\{6\}\{5\}y\; +\; \backslash frac\{5\}\{5\}y\; =\; 150$
$\frac{11}{5}y=150\backslash frac\{11\}\{5\}y\; =\; 150$

Step 3: Solve for $yy$:

Multiply through by 5:
$11y=75011y\; =\; 750$
$y=\frac{750}{11}y\; =\; \backslash frac\{750\}\{11\}$

Step 4: Solve for $xx$:

Substitute $y=\frac{750}{11}y\; =\; \backslash frac\{750\}\{11\}$ into $x=\frac{6}{5}yx\; =\; \backslash frac\{6\}\{5\}y$:
$x=\frac{6}{5}\cdot \frac{750}{11}x\; =\; \backslash frac\{6\}\{5\}\; \backslash cdot\; \backslash frac\{750\}\{11\}$
$x=\frac{4500}{55}x\; =\; \backslash frac\{4500\}\{55\}$
Simplify:
$x=\frac{200}{11}x\; =\; \backslash frac\{200\}\{11\}$

Final Answer:

The greater number is $\frac{200}{11}\backslash frac\{200\}\{11\}$.