Lesson 4: Unknown Numbers in Multiplication

Definition

To solve for an unknown number, use its inverse operation. We can find an unknown factor by dividing the product by the known factor. We find an unknown dividend by multiplying the other two numbers.

What’s next

This is a foundational concept for algebra. Next, you’ll see worked examples showing how to solve for unknowns presented in different formats, including fractions and standard algebraic notation.

Property

In a multiplication problem, an unknown factor can be found by dividing the product by the known factor. If $a \times b = c$, then $a = c \div b$ and $b = c \div a$.

Examples

• To solve for $x$ in $8x = 96$, divide the product (96) by the known factor (8). $96 \div 8 = 12$, so $x = 12$.
• Find the value of B in the problem: $\begin{array}{r} 20 \\ \times \quad B \\ \hline 500 \end{array}$. Divide 500 by 20 to get $B=25$.
• If a number times 9 equals 108, find the number. $108 \div 9 = 12$. The unknown number is 12.

Explanation

Think of multiplication and division as superhero and sidekick—they are inverse operations that undo each other! If a factor mysteriously vanishes from your equation, just use division to unmask it. You take the final result (the product) and divide by the factor you still have to find the missing one.

Property

In a division problem, the dividend can be found by multiplying the divisor and the quotient. If $k \div a = b$ or $\frac{k}{a} = b$, then $k = a \times b$.

Examples

• To solve for $k$ in $\frac{k}{5} = 20$, multiply the divisor (5) and the quotient (20). $5 \times 20 = 100$, so $k = 100$.
• What number divided by 8 gives you 12? Multiply $8 \times 12 = 96$. The dividend is 96.

Explanation

The dividend is the big number getting split up. If it goes missing, you can rebuild it! Just multiply the other two parts—the divisor (what you divide by) and the quotient (the answer)—together. It's like putting the pieces of a puzzle back together to see the original picture.

Property

In a division problem, an unknown divisor can be found by dividing the dividend by the quotient. If $c \div m = a$, then $m = c \div a$.

Examples

• To solve for $m$ in $180 \div m = 9$, divide the dividend (180) by the quotient (9). $180 \div 9 = 20$, so $m=20$.

• Find the value of $D$ in the problem $D \overline{)169}^{13}$. This is the same as $169 \div D = 13$. So, divide $169 \div 13$ to get $D=13$.