Sample quiz on word problems using linear systems
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1. Let $x$ and $y$ be two numbers whose sum is $12$ and whose difference is $4$. If $x$ is the smaller number, the correct linear system is $\cdots$?
   
   $x+y=12,~x-y=4$
   $x-y=12,~x+y=4$
   $x+y=12,~y-x=4$
   $x-y=12,~x+y=-4$
   
2. $32$ is to be divided into two parts, $x$ and $y$. If $x=\frac{1}{4}y$, the correct linear system is $\cdots$?
   
   $x+y=32,~x=\frac{1}{4}y$
   $x-y=32,~x=\frac{1}{4}y$
   $y=32x,~x=\frac{1}{4}y$
   $x=32y,~x=\frac{1}{4}y$
3. Find two numbers whose sum is $13$ and whose difference is $5$.
   
   $7$ and $6$
   $9$ and $4$
   $5$ and $8$
   $10$ and $3$
4. Two numbers add up to $18$. Twice the first number added to the second number gives $27$. The numbers are $\cdots$
   
   $8$ and $10$
   $9$ and $9$
   $11$ and $7$
   $12$ and $6$
5. Divide $40$ into two parts, so that one part is two-thirds of the other.
   
   $10,30$
   $12,28$
   $16,24$
   $15,25$
6. Divide $100$ into two parts, so that one part is $\frac{9}{16}$ of the other
   
   $36,64$
   $18,82$
   $30,70$
   $28,72$
7. The digits of a two-digit number add up to $7$. If the digits are reversed, the number increases by $9$. Find the original number.
   
   $43$
   $25$
   $34$
   $52$
8. The digits of a two-digit number add up to $10$. If the digits are reversed, the number decreases by $36$. What is the original number?
   
   $37$
   $46$
   $64$
   $73$
9. A man has $\$5$ notes and $\$10$ notes in his pocket. Altogether, he has $50$ notes that amount to $\$450$. How many of each denomination does he have?
   
   $20\times \$5$ and $30\times\$10$
   $10\times\$5$ and $40\times \$10$
   $40\times\$5$ and $10\times\$5$
   $25\times\$5$ and $25\times\$10$
10. Divide $72$ into two parts, so that one part is $50\%$ of the other.
    
    $28,56$
    $24,48$
    $32,64$
    $34,68$