A two-digit number divided by another two-digit number gives 4.375. What are the numbers
The Correct Answer and Explanation is:
Let’s solve the problem step by step:
We are given that a two-digit number divided by another two-digit number equals 4.375.
Step 1: Convert 4.375 to a fraction
We start by expressing 4.375 as a fraction:4.375=4+0.375=3584.375 = 4 + 0.375 = \frac{35}{8}4.375=4+0.375=835
So the quotient is 358\frac{35}{8}835, meaning the division can be expressed as:NumeratorDenominator=358\frac{\text{Numerator}}{\text{Denominator}} = \frac{35}{8}DenominatorNumerator=835
This tells us that the two-digit numbers involved must be in the same ratio as 35 to 8. If we scale both numerator and denominator by the same factor, the ratio remains unchanged.
Step 2: Multiply to reach two-digit numbers
Let’s try multiplying the numerator and denominator of 358\frac{35}{8}835 by integers to find values that are both two-digit numbers.
Start with multiplying both by 2:35×28×2=7016(16 is not two-digit)\frac{35 \times 2}{8 \times 2} = \frac{70}{16} \quad (\text{16 is not two-digit})8×235×2=1670(16 is not two-digit)
Try 3:\frac{105}{24} \quad (105 is three-digit}
Try multiplying both by 1:358(8isnottwo−digit)\frac{35}{8} \quad (8 is not two-digit)835(8isnottwo−digit)
Try multiplying by 2.5:35×2.5=87.5(Not an integer)35 \times 2.5 = 87.5 \quad (\text{Not an integer})35×2.5=87.5(Not an integer)
Instead, let’s check the exact values:xy=4.375⇒x=4.375⋅y\frac{x}{y} = 4.375 \Rightarrow x = 4.375 \cdot yyx=4.375⇒x=4.375⋅y
Let’s try different two-digit values for yyy and see when xxx becomes a two-digit integer.
Try y=8y = 8y=8:
x=4.375×8=35x = 4.375 \times 8 = 35x=4.375×8=35 → 35 is valid, but 8 is not a two-digit number.
Try y=16y = 16y=16:
x=4.375×16=70x = 4.375 \times 16 = 70x=4.375×16=70 → Both 70 and 16 are two-digit numbers.
So, the two-digit numbers are:
70 and 16
Final Answer:
70 ÷ 16 = 4.375
Explanation
To find two-digit numbers that satisfy the condition where one divided by the other equals 4.375, we first convert the decimal 4.375 into a fraction. This conversion helps simplify the problem and makes it easier to identify values that hold the same ratio. The decimal 4.375 can be broken down into 4 plus 0.375. We know 0.375 is equal to 3/8, so the entire number becomes 4 + 3/8, which equals 35/8.
This means that the result of the division is the fraction 35 over 8. So we are essentially searching for two two-digit numbers that simplify down to 35 over 8. In mathematics, if a fraction is simplified, any multiple of the numerator and the same multiple of the denominator will still equal the original value. Therefore, we can multiply both the numerator and denominator of 35/8 by the same whole number to reach a pair of two-digit numbers.
By trying a few small multipliers, we find that multiplying both the numerator and denominator by 2 gives us 70 and 16. These are both two-digit numbers. When we divide 70 by 16, we get exactly 4.375, confirming that these are the numbers we are looking for.
Hence, the two-digit numbers that generate 4.375 when divided are 70 and 16.
